Detecting distributional differences between two independent samples is a fundamental problem in statistics and machine learning. Nonparametric two-sample testing provides a principled framework for determining whether two samples are drawn from the same underlying distribution, without assuming any specific parametric form for the distribution. In this study, we propose a new two-sample test statistic based on a newly introduced integral probability metric (IPM), using a specially designed parametric discriminator class with a single node of a neural network. We show that the resulting test statistic, called PReLU-IPM, is nonparametric and establish theoretical guarantees for the associated two-sample testing procedure, PReLU-TST, including its consistency and asymptotical equivalence to nonparametric IPM-based tests under regularity conditions. By analyzing multiple simulated and real benchmark datasets, we demonstrate that PReLU-TST achieves higher power across a range of alternatives or performs comparably to its competitors, for finite samples.

We study the statistical and computational limits of learning bounded linear operators between Sobolev spaces from noisy input-output data. In wavelet coordinates, the problem is recast as an infinite-dimensional matrix regression problem with a heterogeneous two-sided multiscale structure. We establish minimax rates under Sobolev operator-norm loss and construct a finite-resolution blockwise least-squares estimator attaining these rates. The analysis reveals a nonuniform local estimation difficulty across scales, which can be exploited algorithmically: by assigning scale-adaptive sample sizes, the estimator achieves the optimal computational cost among dense least-squares implementations.

Quantitative evidence synthesis method has become a central tool for integration of findings across multiple studies, multi-centre trials, and multi-source cohort data. However, the identification and interpretation of non-replicable, outlying, and influential studies remain insufficiently addressed in practice, despite their potential to substantially affect the robustness and credibility of meta-analytic conclusions. In this paper, we clarify the conceptual distinctions between non-replicability, statistical outlyingness, and study influence, emphasizing that these concepts are related but not interchangeable. We then review the standard principles and procedures of model diagnostics for detecting outlying and influential studies in meta-analysis, together with their underlying statistical rationale. Building on recent methodological developments, we further discuss several practical and methodological refinements, including approaches for handling imprecise and correlated sampling variances, robust diagnostic procedures, and graphical tools for facilitating the identification and interpretation of unusual studies. Finally, we summarize recent advances in outlier and influence diagnostics and provide recommendations for the cautious interpretation and evaluation of studies identified as potentially non-replicable, outlying, or influential within meta-analytic frameworks.

We give a characteristic-function formulation of Kunchenko's stochastic-polynomial construction for settings in which raw moments may fail to exist. In the finite-variance trigonometric case, the coefficients of the Kunchenko normal system are expressed through the characteristic function and its derivative. In the moment-free case, empirical characteristic functions on a fixed finite frequency grid define a bounded discrepancy geometry that remains meaningful for Cauchy, symmetric stable, and other heavy-tailed laws. We prove well-definedness and finite-grid almost sure consistency of this empirical characteristic-function geometry. We introduce the associated minimum-CF-distance estimator and establish its identifiability, strong consistency, and asymptotic normality on a fixed grid, with a covariance built from bounded trigonometric moments that stays finite even for Cauchy and stable laws; refining the grid increases the optimal-weight information monotonically to the Fisher information, so the estimator is asymptotically efficient in the dense-grid limit. We also relate bounded sine scores to weak stochastic-polynomial estimating equations. A small Lean 4 / Mathlib supplement checks selected deterministic identities underlying the bounded-score construction; convergence arguments and statistical interpretation remain outside the formalization.

We study an open problem of understanding the effects of the minimum component separation on the convergence rates of parameter estimation in finite Gaussian mixtures. We address this by developing a unified geometric framework based on novel Hellinger lower bounds that directly relate discrepancies between mixture densities directly to Wasserstein distances between their underlying mixing measures, with explicit dependence on both the minimum separation and the minimum weight. Our approach combines carefully designed interpolation polynomials with confluent divided difference techniques to construct specialized moment-extraction test functions. When the number of components is known, these bounds uncover a localization phenomenon: the separation complexity is driven strictly by the spatial configuration of mixture components, namely, whether they are concentrated in a single cluster, partitioned into multiple clusters separated by a macroscopic gap, or arranged without any structural constraints. On the other hand, when the number of components becomes unknown and is over-specified, the separation complexity is slightly reduced, while the minimum mixture weight disappears entirely from the convergence rates due to a transition from first-order to second-order Wasserstein geometry. As a consequence, we obtain separation-dependent convergence rates that continuously interpolate between point-wise and uniform estimation regimes, thereby settling the fundamental limits of parameter recovery in finite Gaussian mixtures.

Functional regressors complicate inference in linear regression problems so that the bootstrap can play a useful role in quantifying uncertainty and calibrating intervals. The best bootstrap in practice, though, can depend on factors in the data as well as computational considerations and existing bootstraps can have limitations: residual bootstrap is computationally fast and simple but may fail when the errors are heterogeneous, while paired bootstrap applies more generally in functional linear regression at a cost of much higher computation. To bridge this gap, we develop a wild bootstrap method for functional linear regression, which is akin to a modified version of residual bootstrap but designed to have a wide scope of application like paired bootstrap, including to heteroscedastic errors. Its theoretical consistency is established and numerical studies suggest that wild bootstrap can provide accurate and computationally fast inference. Importantly, we also suggest a practical and effective approach of selecting truncation levels, specifically designed for mean response inference problems. The proposed bootstrap in functional linear regression is further illustrated through a weather data example, and an accompanying R package BTSinFLRM provides numerical implementations.

Generalized estimating equations (GEE) are widely used for population-averaged inference on correlated binary responses, but ordinary GEE can fail under separation, a situation that is more likely in small-sample, sparse, or rare-event settings, leading to nonconvergence, infinite or extreme estimates, and unreliable inference. Existing penalized GEE (PGEE) approaches mitigate some of these problems but do not generally guarantee finite estimates under nonindependence working structures and often rely on correlation-coefficient parameterizations whose admissible range shrinks as fitted probabilities approach zero or one, forcing the working association toward independence under separation. We propose a PGEE framework that combines a Jeffreys-prior penalty with marginalized odds-ratio working parameterizations. The odds-ratio parameterization avoids this failure, while the penalty, with tunable strength $δ$ and default $δ= 1/2$, stabilizes estimation under separation. Under working independence, PGEE reduces to the Jeffreys-prior penalized maximum-likelihood estimator, yielding finite estimates for logit, probit, complementary log-log, and cauchit links. Under nonindependence odds-ratio structures, where a formal finiteness guarantee is unavailable, PGEE achieves near-complete empirical convergence even in separated settings. We also propose one-step and hybrid variants, OPGEE and HPGEE, that reduce computational cost. Simulations show that all three variants substantially outperform ordinary GEE under separation while retaining the performance of ordinary GEE in regular settings. We illustrate the method using a respiratory-illness trial in which ordinary GEE fails, and provide an implementation in the R package geer.

Generalized estimating equations (GEE) are widely used for correlated data, but with small to moderate numbers of independent clusters the ordinary GEE regression estimators can be substantially biased. We develop a first-order bias-reduction principle for GEE by viewing the estimator as a clustered-data $M$-estimator and deriving an adjustment to the estimating equations that targets the leading bias term while accounting for the dependence of the working covariance on the mean parameters. The resulting class includes three bias-reduced estimators and three one-step bias-corrected analogs, nesting the bias-corrected estimator of Lunardon and Scharfstein (2017) and the bias-reduced and bias-corrected estimators of Paul and Zhang (2014) as special cases. The framework applies to general response types through correlation-coefficient parameterizations for the association structure and extends to correlated binary data through pairwise odds-ratio parameterizations, yielding the first bias-reduced and bias-corrected GEE estimators under this parameterization, for which the marginal-mean compatibility constraints are far less restrictive than those of correlation-coefficient parameterizations, making them better suited for small-sample settings. Under standard regularity conditions, all six estimators share the same asymptotic distribution as the ordinary GEE. Simulation studies show that the proposed estimators reduce bias while maintaining efficiency and coverage close to those of ordinary GEE across a range of settings, and a clinical trial analysis illustrates the proposed estimators in practice. Software is available in the R package geer.

Interpreting machine-learning models has attracted increasing attention, particularly in the physical sciences, where one often seeks to understand the underlying mechanisms rather than merely make predictions. Multiple linear regression is often regarded as an interpretable alternative to more complex models, such as deep neural networks, because its predictions are expressed as explicit weighted sums of input features. However, when input features are strongly correlated, namely in the presence of multicollinearity, the learned weights can exhibit large dataset-to-dataset fluctuations and oscillatory behavior across physically similar features, making their interpretation difficult or even impossible. Although the instability of the weights under multicollinearity is well known in statistics, its consequences for physical interpretation, in particular its connection to oscillatory weights across physically similar features, have not been systematically clarified. Here, we theoretically discuss the mechanism behind this loss of interpretability by analyzing the eigenmodes of the feature correlation matrix. We show that small-eigenvalue modes associated with multicollinearity amplify fluctuations in the weights and generate oscillatory patterns that do not necessarily reflect meaningful contributions. We test this theoretical picture numerically on physics datasets and show that Ridge regularization suppresses these unstable modes, although the resulting weights must still be interpreted with caution. We further confirm the generality of our findings beyond physics by analyzing a diverse collection of publicly available datasets. Our results clarify why, in the presence of multicollinearity, physical interpretation can remain difficult even for linear regression models.

A single observed network reflects several mechanisms at once: communities, hubs, and clustering coexist in one graph, each a different model. We treat the network as a combination of candidate mechanisms and study, from a single graph, how strongly each mechanism contributes and how they combine. We address two questions. The first is how to measure each mechanism's contribution when the mechanisms must themselves be estimated from the graph: fitting the mechanisms and their strengths from the same data biases the strengths toward zero, and a correction removes this bias and yields valid confidence intervals. The second is whether the rule of combination is itself recoverable: when a graph is generated by two mechanisms acting together, the graph alone determines whether they combine additively or interact, exactly when the graph is dense enough, a sharp threshold below which no test can decide. The estimate calibrates the candidate mechanisms against the observed edges. We establish matching minimax rate, against a known-design benchmark and the estimated-design problem itself, confirm the methods in simulation, and apply them to real networks, where the signed coefficients recover known structure and, in one case, a confidence interval excludes any positive contribution from a candidate mechanism.

Analyzing the mean response of study subjects in psychological research is a standard, well-justified practice. However, theoretical arguments and empirical evidence also suggest that there is value in investigating other aspects of the distribution of such responses, such as their variability or skewness. A particular challenge that practitioners face is statistical modeling of associations between distributional features and other outcomes of interest. The most common approach is to perform estimation in two steps: distributional features are estimated first, and then those estimates are used as predictors for the relevant outcomes. Such an approach is most amenable to implementation in standard statistical software, but it ignores estimation error and can therefore lead to biased estimates and increased error rates. We introduce Distributional Feature Latent Variable Models (DFLVM), a general framework that represents between-person difference in distributional features as random intercepts. These intercepts can be simultaneously used as predictors for downstream outcomes and their associations estimated in a single estimation step. We compare the performance of our approach against two-step procedures in a simulation study and through a re-analysis of a real dataset.

Kernel density estimation depends largely on one choice, the smoothing bandwidth. We treat bandwidth selection and density estimation in the characteristic-function domain, where the cyclic group-averaged covariance of the binned data has the squared empirical characteristic function as its spectrum: the true characteristic function sits over a sampling-noise floor of $1/n$, and the bandwidth is the spectral cutoff where the two meet. Several methods follow. An automatic selector strips the floor and minimizes a frequency-domain error criterion, matching the rule of thumb on smooth densities and approaching the best fixed bandwidth on multimodal ones. An adaptive estimator generalizes the fixed kernel to the per-frequency optimal Wiener taper, matching or surpassing the best fixed bandwidth on most standard densities, including sharply peaked and comb-like cases where fixed bandwidths fail; deconvolution under known measurement error follows in the same domain. Because the Wiener estimator resolves sharp structure but does not fit smooth bases as economically as a mixture, a Gaussian mixture is combined with it two ways, a piecewise partition and a superposition of a smooth base and a band-limited residual, the default. A data-driven floor read from the spectrum replaces the assumed $1/n$ floor and stays robust on heaped and rounded data. On the Marron-Wand benchmark scored by exact integrated squared error, the advantage emerges with sample size, a bias-variance tradeoff: the spectral estimators carry low bias but pay in variance, so cross-validation leads at $n=100$ while the Wiener filter and superposition take the top two ranks at $n=5000$. The methods are validated on six real datasets (CRSP returns, NHANES self-reports, CMS dimuon and SDSS spectra, a random-beacon stream, and UNSW-NB15 traffic) and on a synthetic-data quality check. All experiments are reproducible.

Inferring the strength of conditional dependence and testing conditional independence are fundamental problems in statistics. A recent breakthrough by Azadkia and Chatterjee introduced, for the first time, a conditional dependence measure that equals $0$ if and only if the variables under study are conditionally independent, and equals $1$ if and only if they are conditionally perfectly dependent. They further proposed a computationally efficient and strongly consistent estimator, $T_n$, based on an ingenious use of ranks and nearest neighbors. Despite these attractive features, the asymptotic theory of $T_n$ has remained largely undeveloped. This paper closes that gap. We prove that, under general dependence, $T_n$ is asymptotically normal and its limiting variance admits a closed form. We also construct consistent variance estimators that are computationally efficient and implementable in $O(n\log n)$ time. Taken together with existing bias-correction methods, these results provide a complete inferential theory for $T_n$.

Scientific discovery relies on large-scale hypothesis testing. However, the capacity to identify true discoveries while controlling false discovery faces major challenges: obtaining relevant reference data (the null distribution) is resource-intensive, leaving finite-data uncertainty, and the procedure should account for the inherent structure in the hypothesis space, when such structure exists. Here, we present a framework for controlling the false discovery rate both when each hypothesis is evidenced only by a finite count of null draws, leaving its p-value uncertain, and when the hypothesis space carries arbitrary structure, requiring only that the structure be represented through a suitable reproducing kernel. We present two decision rules that are both robust to structural mis-specification, yet offer a distinct trade-off between exact FDR control and statistical power. The first rule guarantees exact FDR control; the second maximizes power by adapting mirror-statistic control into count space, utilizing an analytical framework to assess FDR control when exact mirror symmetry is relaxed. Furthermore, the tractability gained by the RKHS framework allows us to directly investigate finite-data uncertainties, which we leverage to suggest a policy for the efficient allocation of null distribution samples.

State-of-the-art approaches to magnetic anomaly detection rely on the generalized likelihood ratio test (GLRT). These approaches are based on the formulation of a parametric model of the source to be detected, expressed in a suitable functional basis. One of the primary objectives of this study is to demonstrate that, for a given measurement configuration, the signal is constrained to evolve within a restricted subset of the space generated by these functional bases. The parametric representation of the signal is identified as a semi-algebraic space which, for the dipole model used in this article, turns out to be a cone outside of which the estimated signal does not satisfy the physical equations. Thus, a second objective is to exploit this property to constrain the signal parameters in the GLRT to belong to the semi-algebraic space, in order to improve detection performance. The performance gain of the proposed algorithm is compared to the one of conventional approaches; numerical simulations show that the proposed approach not only outperforms state-of-the-art methods but can even provide results close to those of the clear-seeing (optimal) receiver.

Ensemble classifiers are predictive models that combine the results of simpler base models, often by majority vote. A classic example is random forests, which combine the predictions of decision trees. Ensembles that use more base models can be more accurate but also more costly to train and run. In this paper, we consider strategies for reducing the computational cost of binary classification using an approach from the field of sequential testing. Rather than evaluating all the base models and taking a majority vote, we evaluate the base models sequentially and stop execution when a clear majority emerges. We consider three different notions of optimality for early-stopping strategies that minimize the number of base models executed while controlling the rate of disagreement with the full ensemble. For each notion of optimality and allowable disagreement rate, we show that a linear program can be constructed and solved efficiently to find the optimal stopping strategy. We tested these methods on real-world datasets taken from the UC Irvine Machine Learning repository, and on the benchmark datasets proposed by Grinsztajn et al. We found that on most datasets, these methods provide speed-ups of 4x or more while controlling disagreement at 0.1%

Comparing group means is foundational to many statistical areas, including two-sample studies, randomized trials, and difference-in-differences designs, yet heavy-tailed outcomes can make conventional estimators unstable. A common remedy is to winsorize the data before estimating the target mean contrast. The dominant approach, pooled winsorization, computes winsorization thresholds from the combined sample across all groups, while the rarely used alternative, separate winsorization, computes them within each group. We study finite-sample deviation bounds for these two winsorization strategies, and we prove an impossibility result: no deterministic rule for selecting the pooled winsorization level can attain the sub-Gaussian rate. In contrast, separate winsorization attains this rate, and the guarantee extends to general linear contrasts of group means. Simulation studies confirm that pooled winsorization can have substantial bias, while separate winsorization remains nearly unbiased and concentrates well around the truth. These results support a simple recommendation: winsorize within each group rather than after pooling.

The comparison of methods of measurement is a common problem in clinical practice; as novel methods are developed, establishing their agreement with existing methods is crucial. The probability of agreement (PoA) has previously been proposed as an intuitive and informative means of assessing agreement between two methods of measurement. It straightforwardly quantifies the likelihood that two measurements by different methods on the same subject are clinically indistinguishable. In this paper, we overhaul and extend the PoA methodology by developing an inference framework that relaxes several restrictive assumptions made in previous implementations, ultimately increasing its utility in a wider range of applications. We illustrate this more flexible methodology in an example that compares methods of measuring total Prostatic Specific Antigen (tPSA). And we thoroughly investigate its performance via simulation. This work dramatically increases the flexibility, availability, and hence impact of the PoA approach for method comparison.

Meta-analyses are often based on too few studies to justify the asymptotic methods underlying the statistical procedures applied to them. We consider higher-order asymptotics as a remedy. We derive the Bartlett correction for the idealized Gaussian case, correcting the formula currently appearing in the literature.

Density estimation in high-dimensional settings is an important and challenging statistical problem.Traditional methods based on kernel smoothing are inefficient in high dimensions due to the difficulties in specifying appropriate location-adaptive kernels. In this work, we introduce pre-training, a key idea behind many cutting-edge AI technologies, to the context of non-parametric density estimation. By establishing a pre-trained neural network that can recommend an appropriate location-adaptive kernel for each sample point, efficient density estimation with adaptive kernels is achieved in high dimensions. A wide range of numerical experiments show that this strategy is highly effective for improving density-estimation accuracy, when the target distribution is close to the distribution family for pre-training. When the target distribution is substantially different from the pre-training distribution family, the benefit from the proposed pre-training strategy may be diluted, but can be reactivated by an additional fine-tuning procedure.

We prove that any random variable $X$ whose moment generating function is point-wise upper bounded by that of $ G \sim \mathcal{N}(0,1) $ must be dominated by $ G/\mathbb{E}[|G|] $ in convex order, meaning $ \mathbb{E}[f(X)] \le \mathbb{E}[f(G/\mathbb{E}[|G|])] $ for all convex $f$. This is sharp as witnessed by $ X \sim \mathrm{Unif}(\{-1,1\}) $ and $ f(x) = |x| $.

We revisit size controllability results in Pötscher and Preinerstorfer (2025) concerning heteroskedasticity robust test statistics in regression models. For the special, but important, case of testing a single restriction (e.g., a zero restriction on a single coefficient), we povide a necessary and sufficient condition for size controllability, whereas the condition in Pötscher and Preinerstorfer (2025) is, in general, only sufficient (even in the case of testing a single restriction).

We study one-sided and $α$-correct sequential hypothesis testing for data generated by an ergodic, finite-state Markov chain. The null hypothesis is that the unknown transition matrix belongs to a prescribed set $P$ of stochastic matrices, and the alternative corresponds to a disjoint set $Q$. We establish a non-asymptotic instance-dependent lower bound on the expected stopping time of any valid sequential test under the alternative, which is asymptotically tight. Our novel analysis improves the existing lower bounds, which are either asymptotic or provably sub-optimal in this setting. Our lower bound incorporates both the stationary distribution and the transition structure induced by the unknown Markov chain. We further propose an optimal test whose expected stopping time matches this lower bound asymptotically as $α\to 0$. We illustrate the usefulness of our framework through applications to sequential detection of model misspecification in Markov Chain Monte Carlo and to testing structural properties, such as the linearity of transition dynamics, in Markov decision processes. Our findings yield a sharp and general characterization of optimal sequential testing procedures under Markovian dependence.

Identifying brain regions that exhibit altered functional connectivity between cognitive or emotional states is a fundamental problem in neuroscience. We propose SpARCD (Spectral Analysis for Revealing Connectivity Differences), a statistical framework for detecting detecting condition-specific patterns of functional connectivity. SpARCD uses distance correlation, a dependence measure sensitive to both linear and nonlinear associations, to construct weighted region-wise connectivity graphs for each condition. A differential operator obtained through spectral filtering is then used to identify connectivity changes via its leading eigenvectors. To assess statistical significance, we develop a permutation-based testing procedure that yields interpretable region-level significance maps. We establish finite-sample validity of the permutation test and derive asymptotic guarantees for the stability of the resulting region rankings. Simulation studies demonstrate improved power relative to conventional edge-wise and univariate approaches, particularly in settings with nonlinear dependence structures. We applied SpARCD to fMRI data from 113 individuals with early-stage PTSD and 42 controls during emotional and neutral task conditions. The method identified distinct connectivity networks associated with visual processing in both PTSD and control participants. Resting-state comparisons between PTSD and control participants highlighted similar visual networks. SpARCD provides a statistically rigorous and computationally efficient framework for comparing high-dimensional connectivity patterns.

We establish a fundamental connection between optimal structure learning and optimal conditional independence testing by showing that the minimax optimal rate for structure learning problems is determined by the minimax rate for conditional independence testing in these problems. This is accomplished by establishing a general reduction between these two problems in the case of poly-forests, and demonstrated by deriving optimal rates for several examples, including Bernoulli, Gaussian and nonparametric models. Furthermore, we show that the optimal algorithm in these settings is a suitable modification of the PC algorithm. This theoretical finding provides a unified framework for analyzing the statistical complexity of structure learning through the lens of minimax testing.

Standard kernel two-sample tests, such as those based on the Maximum Mean Discrepancy (MMD), aggregate squared differences across all directions in a Reproducing Kernel Hilbert Space (RKHS). However, in finite samples, trailing directional components are noisy, which degrades test power. We propose a novel kernel-based test that resolves this by truncating the spectral decomposition of the MMD, retaining only the well-estimated leading eigen-directions. By aggregating these robust components, our method achieves superior power and robustness, particularly in high-dimensional and unbalanced settings. Furthermore, we introduce a computationally efficient parametric bootstrap procedure for approximating critical values, which is theoretically justified and significantly faster than permutation-based alternatives. Extensive simulations and empirical studies demonstrate that our method maintains strict Type I error control while delivering higher power than existing MMD-based tests.

Let $\mathcal{M}$ denote the class of randomised monotone functions on $\mathbb{R}$ with values in $[0,1]$, and let $U_{\mathcal{M}}\colon \mathbb{R}_+\to \mathbb{R}_+$ be the minimal function for which $$ \mathbb{P}\left\{ \sqrt{η_f}\, \sup_{t\in\mathbb{R}} \left| f_Z(t) - \Exf{f_Z(t)} \right| \ge \varepsilon\sqrt{U_{\mathcal{M}}(η_f)} \right\} \le 2\e^{-2\varepsilon^2} $$ holds for every member $f_Z$ of $\mathcal{M}$ with finite effective sample size $η_f$ and every positive $\varepsilon$. We prove that for every $x> 1$, $$ \left| \sqrt{U_{\mathcal{M}}(x)} - \sqrt{\log_4 x} \right| \le 2 \min\!\left\{ 1,\, \frac{2 \ln(\e + \ln x)}{\sqrt{\ln x}} \right\}\,. $$ The optimal adjustment $\sqrt{U_{\mathcal{M}}(x)}$ matches $\frac{1}{\sqrt{2\ln 2}}\sqrt{\ln x}$ for all $x>1$, with residuals bounded as above.

The estimation of the multivariate normal mean is a fundamental problem, highlighted by the inadmissibility of the MLE for $n\geq 3$ under quadratic loss. While shrinkage and empirical Bayes methods leverage joint structure through geometric reasoning or hierarchical modeling, this paper proposes a class of point estimators derived from the prior-free framework of inferential models. We develop a generalized probability integral transform for independent, non-i.i.d observations, creating a bijective mapping from the sample to an ordered-uniform reference distribution. By combining this bijection with an ordered-uniform predictive random set based on a reweighted Anderson-Darling statistic, we ensure valid and efficient inference that captures the global shape structure revealed by the ordered observations. We further introduce a maximin (bottleneck) criterion for combining multiple plausibility contours. To ensure computability, we develop a sampling-with-replacement surrogate that connects the exact formulation to over-parameterized (g)-modeling. Our approach provides a structural explanation of Stein's paradox, showing that the MLE corresponds to a zero-density point of the joint auxiliary distribution, revealing its implausibility from an auxiliary perspective. Numerical studies show that our estimators are competitive with state-of-the-art auto-modeling methods and outperform classical shrinkage and empirical Bayes methods.

Public AI evaluations are often read as terminal leaderboards, yet the underlying evidence is a selective time series shaped by reporting rules, benchmark revisions, and missingness. Repeated public archives for LiveBench and Open LLM Leaderboard v2 serve as the primary longitudinal record; LMArena provides a preference stress test; and GAIA and tau-bench contribute limited agentic pilots. Together, these archives instantiate a Bayesian inference problem: under a fixed reporting convention, one constructed terminal-only example over $1{,}000$ systems is compatible with two pre-terminal histories, yielding times of $23.03$ or $75.13$ to reach within $0.05$ of the ceiling under the same terminal-tail model. In synthetic posterior comparisons, action-facing diagnostics differ across observation regimes. The candidate selection-aware frontier model fails synthetic recovery, objective-archive prediction, preference transfer, and uncertainty calibration; correspondingly, fixed audit gates reject its stronger claims. An archive-and-adjudication protocol reconstructs public evaluation histories, isolates a verified timing boundary, and falsifies unsupported frontier claims.

Simulation-based inference (SBI) of latent parameters is often hindered by simulator misspecification, the mismatch between simulated and real-world observations caused by inherent modeling simplifications. RoPE, the recent state-of-the-art for robust SBI, addresses this through optimal transport between learned representations of real and simulated observations, but requires ground-truth parameter calibration pairs that are typically unavailable in the very settings where SBI is needed. What practitioners do have is unstructured side-information such as regime labels, instruction text, and policy bulletins. We propose Misspecification-Aware Simulation-Based Inference (MA-SBI), a calibration-free framework that turns this side-channel into a posterior correction. A learned corrector maps side-channel text to an observation-space shift applied before any pre-trained amortized posterior, requiring no retraining and no parameter ground-truth. Our main theorem bounds achievable bias reduction by the mutual information between misspecification and side-channel, with a non-vacuous constant that extends to all sub-Gaussian noise via Donsker-Varadhan. On hide-the-calibration benchmarks, MA-SBI with text alone matches the oracle posterior across 10 seeds and two backbones (TOST equivalence), while RoPE given more data does not. The two approaches are complementary: where misspecification is structural and recoverable from parameter pairs, RoPE dominates, as the theory predicts. A stochastic variant improves posterior-predictive log-likelihood on real COVID and OxCGRT epidemiological data, and correctly leaves the posterior unchanged on a well-specified cognitive-science corpus.

Fairfield and Charman (2022) propose using a Bayes factor to summarize process tracing evidence, but they require researchers to specify the probability of evidence by hand, and this has drawn concern about bias (Zaks 2021). In this paper, we present a solution by deriving such probabilities directly from two fully specified generative models of observation tailored to process-tracing research designs. Our fully specified Bayes factors enable researchers to report how much observation bias a positive conclusion can absorb before flipping in favor of the rival, taking dependence on smoking gun weight into consideration as well. In practice, this means that final conclusions are driven by sensitivity tests more than by Bayes factors themselves. To show the usefulness of our approach we apply the framework to six recent process-tracing studies published in top political science journals.

Engineered robust losses such as Huber, Student-$t$, and generalised cross-entropy make supervised models tolerant of contamination but cannot answer which observations are corrupted. We introduce Neural Bayesian Anomaly Mitigation (NBAM), a general-purpose drop-in loss derived from a Bayesian latent-switch mixture model: the marginal likelihood defines a robust supervised loss, and the associated posterior defines an unsupervised contamination classifier. Like Huber or Student-$t$, NBAM can replace the standard training loss in any supervised pipeline; unlike them, it additionally learns a structured contamination model and returns a calibrated per-sample contamination posterior. A learned input-dependent prior $π_ϕ(x)$ captures the spatial locality of contamination, so that samples near known corruptions are more likely to be flagged, while an Occam penalty emerges automatically and regularises against over-flagging. On CIFAR-10 with asymmetric label contamination, NBAM recovers the structure of the corruption process without supervision: the contamination posterior separates clean from corrupted samples, and the learned anomaly head identifies the direction of every label-flip pair. Alongside these capabilities, NBAM outperforms the four robust-loss baselines considered here at contamination rates 0.2-0.6.

Meta-analysis is a key statistical tool for synthesizing clinical trial data to evaluate treatment effects, yet traditional methods like fixed and random-effects models often fail to handle heterogeneity, study-level covariates, or hierarchical structures effectively. To overcome these limitations, we developed a Bayesian hierarchical meta-analysis framework for robust parameter estimation on small samples and utilized analytical integration for efficient inference. Simulation studies indicated robust estimation of the proposed model. We applied it to the safety profiles of Oxcarbazepine (OXC) and Carbamazepine (CBZ) in epilepsy treatment. The results indicated that OXC was significantly associated with a lower risk of side effects than CBZ. The code and relevant data used in this study are openly available on GitHub at: https://github.com/xsjk/HierarchicalMetaAnalysis.

Longitudinal clinical studies often collect repeated measurements of biomarkers or health-related quality of life together with a time-to-event outcome. These processes are intrinsically linked: longitudinal trajectories may predict event risk, while event occurrence, or its anticipation, can induce informative censoring of the longitudinal process. Joint models provide a principled framework for handling this dependence, but most existing formulations rely on proportional hazards assumptions that may be restrictive and offer limited interpretability on the time scale. We propose a class of semiparametric accelerated failure time joint models that directly model covariate effects on event timing while flexibly capturing longitudinal-event associations. The survival component is specified through an accelerated failure time model with the baseline component represented by a flexible basis expansion, allowing a broad class of smooth baseline specifications. We illustrate the framework using Bernstein polynomial baseline representations and introduce rescaling strategies to improve numerical stability and parameter identifiability under time-warping. Estimation is conducted within a Bayesian framework, enabling joint inference for longitudinal, survival, and association parameters. Simulation studies reflecting realistic longitudinal trajectories, censoring mechanisms, and dependence structures are used to evaluate finite-sample performance. The proposed models show improved recovery of longitudinal treatment effects compared with a standalone linear mixed model when event risk depends on the underlying longitudinal process. Overall, the framework extends existing joint modelling methodology by offering a flexible and interpretable alternative to proportional hazards-based approaches.

Deep neural networks (DNNs) frequently fail to generalize to out-of-distribution (OOD) medical images because of variations in scanners and acquisition protocols. Retraining DNN models to address these distribution shifts is often impractical due to the high cost of acquiring and annotating new medical datasets. To address this, we introduce VarDeepPCA, a novel lightweight variational DNN framework designed to restore/refine degraded segmentation maps by leveraging intrinsic geometric priors. Unlike existing approaches that require target-domain data or extensive pre-training, our VarDeepPCA explicitly learns a distribution of valid anatomical geometries using only small in-distribution (ID) datasets. Theoretically, our novel variational learning framework leverages a reinterpretation of the softmax mapping to implicitly perform exact distribution modeling, thereby enabling computationally efficient, sampling-free learning and inference. This also enables VarDeepPCA to provide uncertainty estimates associated with its restored segmentation maps. We empirically validate our framework across 4 distinct clinical applications, using 14 publicly available datasets, involving segmentation of the myocardium, neuroretinal rim, prostate, and fetal head. Comparisons against 15 existing methods demonstrate that VarDeepPCA consistently restores segmentation maps produced by the existing methods on OOD data to (i) significantly improve anatomical plausibility of geometries and clinical utility of the segmentations, and (ii) significantly reduce errors, without needing any more training data than that used by existing methods.

Online learning has amplified the need to understand how student engagement patterns influence learning outcomes, particularly given the flexibility of technology-mediated environments. To address this, we propose a Bayesian nonparametric dynamic item response theory (IRT) framework that tracks within-individual ability trajectories across instructional units. The proposed model integrates B-spline basis expansions to capture nonlinear effects of engagement behaviors on ability drift, alongside a Mixture-of-Finite-Mixtures (MFM) prior to automatically determine the number of latent learner clusters. This framework overcomes three limitations in the existing literature: (1) rigid linearity assumptions in engagement-ability relationships, (2) dependence on pre-specified cluster counts, and (3) the inability to track longitudinal ability dynamics. We apply the model to longitudinal data from 198 undergraduates completing a 9-chapter introductory statistics course on CourseKata. The model automatically identified four distinct learner profiles: struggling-declining (11\%), low-stable (23\%), mainstream-stable (55\%), and high-improving (12\%). Results indicate that ability trajectories remained remarkably stable across chapters, and engagement quantity metrics did not significantly predict ability drift. These findings suggest that in introductory online statistics education, academic ability primarily reflects a stable pre-existing characteristic rather than a dynamically malleable course outcome. Ultimately, this framework offers a flexible tool for learner profiling to inform adaptive instructional design.

This paper reviews how a diverse set of popular data-driven priors commonly used in Bayesian inverse problems can be unified through their respective score functions. By framing these priors under this common perspective, we show that they can benefit from their straightfoward and effective integration into a recently proposed sampling algorithm. The applicability of this common framework is illustrated by considering several data-driven priors, namely regularization-by-denoising, normalizing flow-based priors, score-based generative models, and convex-ridge regularizers. For these four particular priors, the performance of the method is evaluated when conducting image inpainting and single image super-resolution. These results, as well as those obtained when restoring real images acquired in a geological context, demonstrate the efficiency of the method. This unified framework proves versatile enough to handle any posterior distribution defined by a broad class of score function-based priors, beyond the specific cases considered in this paper.

Sequential Bayesian experimental design is often formulated as a fixed-horizon policy optimization problem, in which the number of experiments is specified before data collection begins. In practical campaigns, however, additional measurements may provide diminishing information relative to their cost, making termination an integral part of experimental design. Common threshold-based stopping rules are easy to implement but myopic, because they compare the current state with a fixed criterion rather than the expected value of future experiments. This work develops a Bayesian optimal stopping framework for sequential experimental design by treating design and stopping as coupled decisions in a finite-horizon sequential decision problem. We prove that, for any fixed design policy, the optimal stopping rule terminates when the immediate terminal reward is no smaller than the expected continuation value. We then derive a policy-gradient method for learning continuous design policies with value-based stopping. The resulting optimization is challenging because the design policy, continuation value, and stopping boundary are mutually dependent, and naïve training can become trapped in early-stopping local optima. To address this difficulty, we introduce a curriculum strategy that gradually transitions from forced continuation to adaptive stopping during training. Numerical studies on a linear-Gaussian benchmark, a nonlinear test case, and a contaminant source detection problem show that the proposed approach learns stable, resource-aware design-stopping policies, with the largest gains in settings with strong sequential dependence.

Inferring the infinitesimal rates of continuous-time Markov chains (CTMCs) is a central challenge in many scientific domains. This task is difficult because the number of rates grows quadratically with the state space, rates can be strongly dependent, and many transitions may be only partially observed. We introduce a Bayesian framework that models CTMC rates as flexible functions of covariates through Gaussian processes. This enables nonlinear covariate effects, improves inference by incorporating external information, and helps identify potential drivers of CTMC dynamics. For posterior inference, we use Hamiltonian Monte Carlo and develop scalable exact and approximate gradients for likelihoods involving repeated matrix exponentials. With $N$ observations and $K$ CTMC states, these gradients reduce the dominant cost of existing derivative calculations from $O(NK^3)$, with large constants, to $O(K^3+NK^2)$, with cheaper constants. We demonstrate the method in Bayesian phylogenetic and phylogeographic inference, where CTMCs are central, and show strong performance on synthetic and real datasets, including empirical quadratic scaling in $K$ even when $N<K$.

We argue for the use of separate exchangeability as a modeling principle in Bayesian nonparametric (BNP) inference. Separate exchangeability is \emph{de facto} widely applied in the Bayesian parametric case, e.g., it naturally arises in simple mixed models. However, while in some areas, such as random graphs, separate and (closely related) joint exchangeable models are widely used, they are curiously underused for several other applications in BNP. We briefly review the definition of separate exchangeability, focusing on the implications of such a definition in Bayesian modeling. We then discuss two tractable classes of models that implement separate exchangeability, which are the natural counterparts of familiar partially exchangeable BNP models. The first is nested random partitions for a data matrix, defining a partition of columns and nested partitions of rows, nested within column clusters. Many recent models for nested partitions implement partially exchangeable models related to variations of the well-known nested Dirichlet process. We argue that inference under such models in some cases ignores important features of the experimental setup. We obtain the separately exchangeable counterpart of such partially exchangeable partition structures. The second class is about setting up separately exchangeable priors for a nonparametric regression model when multiple sets of experimental units are involved. We highlight how a Dirichlet process mixture of linear models, known as ANOVA DDP, can naturally implement separate exchangeability in such regression problems. Finally, we illustrate how to perform inference under such models in two real data examples.

Defining and estimating causal effects for ordinal data is challenging. Standard average treatment effects are not appropriate for ordinal scales, and alternative estimands, such as the probabilities that the treatment outcome exceeds or does not worsen the control outcome, are generally not identifiable. Existing work provides sharp bounds for these quantities based only on marginal distributions. Motivated by a previous observation showing that bounds obtained under an independence working assumption can be substantially tighter, we investigate conditions under which such bounds are valid. We show that commonly used notions of positive dependence, including positive quadrant dependence and positive regression dependence, are not sufficient to justify these bounds. We then propose a new dependence condition, diagonal tail dominance (DTD), under which the independence-based bounds are guaranteed to hold. We explain why this condition is quite strong and may not be appropriate in many settings, limiting the justification for using the independence-based bounds. However, local DTD may be plausible in many applications, and we derive improved bounds that exploit an independence working assumption on selected parts of the probability table. Through theoretical results, numerical examples, and an analysis of data from a clinical trial of a new treatment for acute ischemic stroke, we illustrate the properties of the bounds and the role of the proposed conditions.

An artificial intelligence must have a model of its environment that is causal, supporting reasoning about interventions and counterfactuals, and also combinatorial, supporting generalization to unseen combinations of objects. In this work, we formally study when and how such a model can be learned. We develop relational structural causal models, extending structural causal models (Pearl 2009) to settings where objects and their relations vary. First, we show how answers to not only causal but also observational queries about unseen combinations of objects can not be identified without further assumptions. To enable such identification--including in the presence of unobserved confounding--we define relational causal graphs and derive symbolic identification criteria. Finally, we propose relational neural causal models, a provably correct approach that outperforms non-relational baselines on simulated traffic scenes with varying cars, signals, and pedestrians.

Estimating causal effects with multivariate continuous exposures is challenging because causal exposure-response surfaces can be high-dimensional, complicating estimation and interpretation of joint exposure effects. Such settings arise in environmental epidemiology, where interest centers on the health effects of chemical and pollutant mixtures. We develop causal sufficient dimension reduction (CSDR), a semiparametric framework for representing causal exposure-response surfaces through low-dimensional exposure summaries. We formalize the reduction target as the causal central mean subspace and propose a modular two-stage estimator that decouples nuisance-function estimation from subspace estimation, simplifying implementation relative to existing marginal structural model-based approaches. The reduced exposure preserves the information needed to characterize joint causal effects while enabling efficient downstream estimation. We establish a convergence rate for causal subspace recovery accounting for first-stage nuisance estimation error, show that the structural dimension can be estimated consistently, and introduce a subspace importance score that quantifies the contribution of each exposure to the reduction. In simulations, CSDR yielded more accurate estimation and uncertainty quantification of the exposure-response surface than methods using noncausal dimension reduction or the original exposure. We apply CSDR to study the effect of maternal exposure to PFAS chemical mixtures on infant birth weight in the Atlanta African American Maternal-Child Cohort.

In data centers, tasks are dispatched to various servers to evenly distribute the workload. When a data center considers implementing a new scheduling algorithm, it typically conducts an A/B test prior to deployment to assess the real-world impact of this new method. However, a straightforward A/B test might be interfered with so-called ``Markovian'' interference. We utilized the Differences-in-Q estimator, as developed by Farias et al. (2022), and introduced mixed Differences-in-Q estimators grounded in Little's Law. We show that our A/B testing methods significantly reduce bias and variance when testing various scheduling policies. Extensive simulations were conducted under scenarios like non-stationary arrival rates, heterogeneous service rates, and communication delays. These simulations highlight the robustness and efficacy of our A/B testing approach.

Experimental evaluations of public policy often randomize an intervention within many sites or blocks. Once an overall effect is reported, the question that matters for action is where it occurred. Standard multiple-testing corrections answer with little power because they ignore how the experiment is organized: blocks nest within cohorts, sites, and districts. We organize the hypotheses as a tree that follows this administrative structure and test them top-down, descending into a branch only when its parent null is rejected. We show that stopping rule and valid node-level tests suffice for weak control of the family-wise error rate (FWER). Whether the same procedure also controls the FWER in the strong sense depends on a single quantity computable before any data are seen: an error load that summarizes how rejection probability accumulates along paths through the tree. This diagnostic tells an analyst in advance, from design quantities alone, whether the unadjusted procedure controls the FWER or an adjustment is required. Across 25 block-randomized MDRC education trials it indicates that no adjustment is needed in every one, so the two conditions alone control the FWER while each test runs at the full nominal level; the top-down procedure detects individual blocks that the Hommel correction misses and locates higher-level groups of blocks that bottom-up testing cannot evaluate. For high-error-load designs we derive an adaptive alpha-schedule, prove it controls the FWER on regular, irregular, and pruned trees, and confirm it in simulation. The same diagnostic flags when it is needed: in a design calibrated to the National Job Corps Study, a wide multisite trial of about one hundred centers, the unadjusted procedure inflates the FWER, the adaptive schedule restores control, and top-down testing still detects more affected sites than bottom-up or hierarchical corrections.

Graph Laplacians encode graph structures in matrix form, and thus facilitate the application of linear algebra to graph theory. In statistics, two related families of probabilistic graphical models can be parameterized by graph Laplacians. The first one is the Laplacian-constrained Gaussian graphical model (LCGGM), which imposes that the (pseudo-)inverse covariance matrix of a Gaussian random vector is a Laplacian matrix. Applications include graph signal processing and network topology learning. The second one is the Hüsler-Reiss graphical model, which is considered as an extremal analog of the Gaussian graphical model, and can be used in extremal dependence modeling of floods, heatwaves, and financial losses. For both models, the restriction to positive edge weights in the graph Laplacian gives rise to an approach for graph structure learning that does not require tuning parameters. While these approaches yield a strong model fit in many settings, the resulting graph estimates are typically much denser than the underlying ground truth, limiting interpretability and scalability. In order to improve the accuracy of Laplacian-constrained graph learning, we propose to use spectral graph sparsification as a post-estimation operation. To do so, we replace the original Laplacian estimate by a sparser Laplacian that is spectrally close, and re-fit the model on the resulting graph. We refer to the two resulting methods as Spectral-LCGGM and Spectral-HR. We investigate the properties of the proposed estimators and show several theoretical results on their performance. Furthermore, we demonstrate that the newly proposed methods perform well by running simulations on Erdős-Rényi and stochastic block model graphs, and we also showcase their applications to real data.

This paper proposes a novel nonparametric test to assess the uncorrelatedness between two high-dimensional random vectors. We develop our test by generalizing the random integration proposed by Jiang et al. (2023, 2024), and the resulting test statistic estimates a weighted squared $\mathscr{L}_2$ norm of the covariance matrix. Asymptotic properties of the test statistic are derived by letting both the sample size $n$ and the dimension $p$ diverge to infinity. Under the null hypothesis of uncorrelatedness, our proposed test statistic is asymptotically normal with zero mean and unit variance, without requiring any specification of the relative magnitude regarding $n$ and $p$. Monte Carlo simulations demonstrate the good finite-sample performance of our proposed methods. Compared with many existing tests, our test statistic is more powerful at detecting ``weak but pervasive'' dependence while maintaining a comparable empirical size. The advantages of the proposed methods are further illustrated by an empirical analysis that assesses the correlation between DNA methylation and gene expression.

In high-dimensional semi-supervised linear regression, prediction-powered inference (PPI) corrects an external predictor with a rectifier estimated from the labeled data. In a linear model, however, this rectifier cancels the predictor: PPI and PPI++ reduce to ordinary least squares and can inflate variance when the predictor is close to the oracle. We propose the Debiased External-model-Assisted Lasso (DEAL), which routes the external estimator and the unlabeled covariates into the variance of a debiased estimator, with a bias-aware, cross-fitted shrinkage step that adapts across target-only, near-oracle, and biased-but-informative regimes. We prove coordinate-wise asymptotic normality with an adaptive variance, extend validity to the projection parameter under misspecification and nonlinear labelers, and show that, at a common unlabeled budget, DEAL intervals are shorter than those of debiased Lasso, PPI, and PPI++; a shift-aware variant preserves coverage under covariate shift. In simulations, DEAL intervals are 0.49-0.87 of the debiased-Lasso length, and across six real-data applications spanning astronomy, chemistry, proteomics, and oncology, the last using a large-language-model oracle, they tighten in every case, with median length ratios of 0.23-0.53.

We study the graph alignment problem for correlated Gaussian Orthogonal Ensemble (GOE) matrices, where the goal is to recover a hidden vertex permutation given two correlated symmetric Gaussian matrices $(A, B)$ with correlation $1/\sqrt{1+σ^2}$. While the maximum likelihood estimator is information-theoretically optimal, its computation, which reduces to a quadratic assignment problem, is intractable. Motivated by this, we analyze convex relaxations based on minimizing $\|AX - XB\|_F$ over the set of doubly stochastic matrices and the unit hypercube. We show that when the correlation parameter satisfies $σ= o(n^{-1/2}/\log^4 n)$, the solution of either relaxation $(X^\star)$ concentrates around the ground-truth permutation matrix $(Π^\star)$, i.e., $\|X^\star-Π^\star\|_F^2 = o(n)$, implying recovery of all but a vanishing fraction of vertices after simple post-processing. Combined with existing lower bounds, our results precisely characterize that $\|X^\star-Π^\star\|_F^2$ transitions from $o(n)$ for $σ= \tilde{o}(n^{-1/2})$ to $Ω(n)$ for $σ= \tildeΩ(n^{-1/2})$. In doing so, our analysis significantly tightens prior results and extends them beyond doubly stochastic relaxations.

We study nonparametric change-point detection for high-dimensional data in regimes where inference must be performed from small batches of observations. Our primary focus is the high-dimensional, low sample size (HDLSS) regime, where the sequence length is fixed while the ambient dimension diverges. We propose a dimension-averaged angular kernel scan framework for detecting marginal distributional shifts. The statistic aggregates bounded one-dimensional angular discrepancies across coordinates, yielding a fully nonparametric, hyperparameter-free, and moment-agnostic estimator that remains well-defined without specifying, estimating, or assuming finite marginal moments; for example, under heavy-tailed or contaminated distributions. For the offline single-change problem, we derive an exact population mean factorization into a universal deterministic shape function and a scalar signal factor, and characterize the exact null covariance structure up to a scalar variance factor, both valid for any fixed sample size and dimension. We also establish an HDLSS multivariate central limit theorem under cross-coordinate strong mixing which leads to a variance-calibrated asymptotically distribution-free test, asymptotic type-I error control, and lower bounds on power and localization accuracy. We further extend the offline procedure to a fixed-window sequential monitoring procedure for high-dimensional streaming data, and obtain ARL calibration and worst-case Pollak EDD bounds. Simulation studies demonstrate that the proposed method can accurately detect and localize changes in many challenging HDLSS and streaming high-dimensional settings where moment-based or hyperparameter-sensitive procedures may be extremely unstable or inaccurate.

Debiased inference for high-dimensional regression models has received substantial recent attention to ensure regularized estimators have valid inference. Many existing methods focus on achieving Neyman orthogonality through explicitly constructing projections onto the space of nuisance parameters, which is infeasible when an explicit form of the projection is unavailable. We introduce a general debiasing framework, Debiased Profile $M$-Estimation (DPME), which applies to a broad class of models and does not require model-specific Neyman orthogonalization or projection derivations as in existing methods. Our approach begins with obtaining an initial estimator of the parameters by optimizing a penalized objective function. To correct for the bias introduced by penalization, we construct a one-step estimator using the Newton--Raphson update, applied to the gradient of a profile function defined as the optimal objective function with the parameter of interest held fixed. We use numerical differentiation without requiring explicit calculation of the gradients. The resulting DPME estimator is shown to be asymptotically linear and normally distributed. Through extensive simulations, we demonstrate that the proposed method achieves better coverage rates than existing alternatives with largely reduced computational cost. Finally, we illustrate the utility of our method by applying it to estimate a treatment rule for multiple myeloma.

In this study, an exact coordinate descent algorithm is developed for high-dimensional Huber regression regularized with an elastic net penalty. Unlike existing gradient descent or coordinate descent-type methods, this algorithm remains effective even when the Hessian becomes ill-conditioned due to high correlations between covariates drawn from heavy-tailed distributions. For each coordinate, marginal increments arise solely from inlier observations, while the derivatives remain monotonically increasing over a grid constructed from the partial residuals. Building on conventional coordinate descent frameworks, adaptive variable screening rules are proposed to selectively determine which variables to update at each iteration, thereby accelerating convergence. The convergence of the proposed algorithm is formally analyzed, and practical computational strategies are presented to speed up its execution. These enhancements ensure that the algorithm operates rapidly and stably even under challenging scenarios. Extensive simulation studies involving heavy-tailed noise and highly correlated predictors, along with a real-world data application, demonstrate both the practical efficiency of this method and the benefits of the computational enhancements.

We introduce a new method for neighbourhood selection in linear structural equation models that improves over classical methods such as best subset selection (BSS) and the Lasso. Our method, called KL-BSS, takes advantage of the existence of underlying structure in SEM -- even when this structure is unknown -- and is easily implemented using existing solvers. Under weaker eigenvalue conditions compared to BSS and the Lasso, KL-BSS can provably recover the support of linear models with fewer samples. We establish both the pointwise and minimax sample complexity for support recovery, which KL-BSS obtains. Extensive experiments on both real and simulated data confirm the improvements offered by KL-BSS. While it is well-known that the Lasso encounters difficulties under structured dependencies, it is less well-known that even BSS runs into trouble as well, and can be substantially improved. These results have implications for structure learning in graphical models, which often relies on neighbourhood selection as a subroutine.

The scalar-on-image regression model examines the association between a scalar response and a bivariate function (e.g., images) through the estimation of a bivariate coefficient function. Existing approaches often impose smoothness constraints to control the bias-variance trade-off, and thus prevent overfitting. However, such assumptions can hinder interpretability, especially when only certain regions of an image influence changes in the response. In such a scenario, interpretability can be better captured by imposing sparsity assumptions on the coefficient function. To address this challenge, we propose the Generalized Dantzig Selector, a novel method that jointly enforces sparsity and smoothness on the coefficient function. The proposed approach enhances interpretability by accurately identifying regions with no contribution to the changes of response, while preserving stability in estimation. Extensive simulation studies and real data applications demonstrate that the new method is highly interpretable and achieves notable improvements over existing approaches. Moreover, we rigorously establish non-asymptotic bounds for the estimation error, providing strong theoretical guarantees for the proposed framework.

We propose a computationally efficient algorithm for gradient-based linear dimension reduction and high-dimensional regression. The algorithm initially computes a Mondrian forest and uses this estimator to identify a relevant feature subspace of the inputs from an estimate of the expected gradient outer product (EGOP) of the regression function. In addition, we introduce an iterative approach known as Transformed Iterative Mondrian (TrIM) forest to improve the Mondrian forest estimator by using the EGOP estimate to update the set of features and weights used by the Mondrian partitioning mechanism. We obtain consistency guarantees and convergence rates for estimating the EGOP matrix and the random forest estimator obtained from one iteration of the TrIM algorithm. Lastly, we demonstrate the effectiveness of our proposed algorithm for learning the relevant feature subspace across various settings with both simulated and real data.

Joint prediction sets for multivariate time series should control a single event while adapting to cross-coordinate dependence. We study filtered conformal ellipsoids: a frozen state-space filter emits a one-step predictive mean and covariance, and split-conformal calibration is applied to the resulting Mahalanobis scores. The filter is used to choose the ellipsoid shape; conformal calibration chooses the scalar radius, so the construction benefits from a learned predictive covariance without relying on Gaussian tail probabilities for coverage. The main difficulty is that filtered scores are dependent and learned recurrent filters need not contract in their raw hidden state; we therefore analyse contraction in an observable predictive-law quotient that identifies hidden states producing the same future sequence of emitted Gaussian laws. Under a stable Bayes Gaussian-projection filter, covariance bounds, and a finite-horizon observability Fisher condition, small excess Gaussian negative log-likelihood implies contraction of the learned emitted laws. Combined with a threshold-autocovariance envelope this yields a Chebyshev-type approximate coverage bound for filtered split-conformal prediction under dependence; a sharper Bernstein-type bound requires an additional geometric-mixing concentration assumption. Under Gaussian oracle realisability we also obtain a near-oracle log-volume comparison within the class of conditionally valid Gaussian ellipsoid rules. We instantiate the framework with a GCN-GRU filter with diagonal-plus-low-rank covariance. On moderate-size graph-native traffic benchmarks (METRLA-$20$ and PEMSBAY-$50$), the learned filter gives sharper at-target ellipsoids than static-covariance and non-filter baselines; at full-graph scale and on non-graph-native datasets, factor and copula baselines can be stronger.

We propose a flexible framework for modeling the predictive distributions of nonlinear, possibly multivariate time series. Our approach expresses a general predictive distribution in an appropriate generative representation that is based on a folklore result from measure theoretic probability. This representation provides a direct simulation-based approximation to the predictive distribution, enabling straightforward computation of forecasts for the conditional mean and variance, fan charts, value at risk, expected shortfall, joint tail risks, and other quantities of interest. We estimate this generative representation using a version of conditional generative adversarial networks and provide a formal statistical analysis of estimation under weak temporal dependence. Specifically, estimation is expressed as a particular minimax problem and we establish consistency of its approximate solutions in Hausdorff distance. The empirical relevance of the approach is illustrated using applications to equity returns, realized variance, and realized covariances. The proposed method is also computationally manageable, with estimation in our applications taking approximately one minute on a standard laptop.

We propose a Bayesian framework for joint distributional forecasting of monthly asylum applications across the EU-27. The model decomposes latent application intensities into country-specific random walks and common factors, with idiosyncratic and shared shocks allowed to exhibit heavy tails or stochastic volatility. Using Eurostat data from 2008 to 2026, we evaluate predictive distributions in a rolling out-of-sample exercise, scoring overall distributional accuracy and upper-tail risk. Three findings emerge. First, the preferred specification varies across countries, scoring rules, and horizons, underscoring the need to align models with policy-specific loss functions. Second, joint EU-27 models improve on country-by-country benchmarks, with the largest gains in the upper tail, where preparedness costs are most relevant. Third, random-walk log-intensities provide a useful short-run description of national asylum-application dynamics, especially when combined with flexible innovation dynamics. We conclude by discussing implications for national and EU-level agencies involved in asylum forecasting and preparedness planning.

Conformal prediction provides distribution-free prediction intervals under exchangeability, but many modern data streams are neither independent nor stable. They exhibit recurring regimes, changing seasonal frequencies, abrupt shifts, and gradual drift. We propose drift-aware spectral conformal prediction (DASC), a streaming uncertainty quantification framework for structured non-exchangeable data subject to distributional drift. DASC forms conformal prediction intervals using calibration residuals weighted by local spectral similarity, while a transport-based drift score monitors whether the current test distribution has moved away from past calibration regimes. When drift is mild, DASC borrows calibration residuals from structurally similar historical windows; when drift is severe, it contracts or reweights the calibration pool and updates the target miscoverage level online. The method also reports an effective sample size diagnostic that warns when a weighted conformal quantile is statistically fragile. We establish an approximate coverage bound that decomposes coverage loss into drift, residual mismatch, and weighted effective sample size. In synthetic experiments and five stress-test regimes, DASC maintains near-nominal coverage after drift where rolling, recency-weighted, and spectral-only conformal methods can under-cover. In real electricity and weather streams, DASC reduces average interval width by approximately 28% and 42%, respectively, relative to the best calibrated non-DASC baseline, while preserving calibrated or conservative coverage. A financial volatility example shows a more nuanced regime in which spectral-only calibration is competitive, but DASC retains near-nominal coverage and adds drift diagnostics.

Conformal prediction gives prediction intervals with finite-sample coverage when the data are exchangeable. Many time-indexed datasets are not exchangeable. They have seasons, recurring regimes, changing frequencies, or other forms of structured dependence. This paper studies a simple way to use that structure. We propose spectral adaptive conformal prediction, a method that forms weighted conformal quantiles using local spectral similarity and then updates the target miscoverage level online. The spectral weights choose calibration residuals that look relevant to the current test point. The adaptive update corrects the long-run miss rate when uncertainty changes over time. We give an approximate coverage result for the fixed spectral weighted quantile and a deterministic long-run calibration result for the adaptive update. Simulations with recurring regimes and slowly changing frequencies, together with three U.S. real-data examples, show that the hybrid method can improve on fixed spectral weighting, while also showing that spectral weighting must be monitored through effective sample size diagnostics.

Rare events in time series are critical to model but hard to learn due to data scarcity. Current generative models struggle with extreme values. We observe that rare events leave distinct topological fingerprints - transitions in Betti numbers from point-cloud embeddings - that are more stable and discriminative than statistical moments. We introduce PHINN, a flow-matching framework using dynamic Betti curves as conditioning signals and a persistence landscape loss for homology consistency. It scales to multivariate data, includes a natural-language interface to set Betti targets, supports cross-domain meta-learning and few-shot generation, and provides certified adversarial robustness. On financial, epidemiological, and multi-modal benchmarks, PHINN outperforms statistical and diffusion baselines in topological fidelity (beta-RMSE down 41-63%, transition accuracy up 84%) and matches jump-diffusion models in tail coverage while exceeding them in shape fidelity. All results have 95% confidence intervals.

A system of coupled oscillators provides a fundamental framework for modeling a wide range of physical and biological phenomena. In neuroscience, the central nervous system exhibits synchronized oscillatory activity with adjacent brain regions, giving rise to traveling wave dynamics for instance during sleep. Similarly, in the gastrointestinal system, neuromuscular cells coordinate their oscillations to generate propagating waves of slow wave activity. To estimate probability distributions of multivariate phase relationships, existing approaches typically rely on equilibrium thermodynamics, expressing the system in a Boltzmann form through a pairwise exponential family distribution. However, these assumptions are often violated in real-world systems, which are inherently dynamic and frequently transition between equilibrium and non-equilibrium regimes. To address this, we propose an efficient method for estimating the probability distribution of coupled oscillators that does not assume thermodynamic equilibrium. Using a Langevin dynamics-based construction, the approach enables accurate modeling even in non-equilibrium regimes. The maximum likelihood estimation method is shown to have a closed form algebraic solution in the high sampling rate regime, a condition commonly satisfied by modern data acquisition systems, which makes it readily applicable in practice. We demonstrate its robustness on simulated data, where it outperforms existing approaches in non-equilibrium settings, and further illustrate its utility for characterizing dynamic brain traveling waves in response to brain stimulation and in hypothesis testing within the context of electrophysiologic recordings of the human stomach.

Comparing multivariate yield quality distributions across spatially referenced agricultural fields is complicated by two pervasive features: non-normality and spatial autocorrelation. Classical procedures such as ANOVA, MANOVA, and standard rank tests assume independence and therefore exhibit severe Type I error inflation when spatial dependence is present. We propose a nonparametric spatial Cramer-von Mises-type test based on kernel-smoothed empirical copula processes constructed from pooled componentwise ranks. Spatial kernel weights account explicitly for local dependence, while the rank transformation removes sensitivity to marginal distributional form. Under fixed-domain infill asymptotics and polynomial alpha-mixing conditions, we establish weak convergence of the smoothed empirical copula process to a mean-zero Gaussian limit and show that the resulting quadratic test statistic converges to a weighted sum of chi-squared random variables restricted to the K-1-dimensional contrast subspace. Practical inference is obtained through a Satterthwaite approximation calibrated using the exact discrete spatial covariance operator under a Gaussian copula model. Monte Carlo experiments with bivariate log-normal spatial data demonstrate that the proposed test maintains nominal size across varying strengths of spatial dependence, in contrast to classical parametric and non-spatial rank-based methods, which become severely anti-conservative. The procedure provides a theoretically justified and computationally tractable framework for comparing multivariate spatial yield distributions in precision agriculture and related applied settings.

Comparing yield quality distributions across multiple agricultural fields is fundamental for evaluating management practices, yet it is complicated by two pervasive data characteristics: non-normality and spatial autocorrelation. Traditional parametric tests, such as ANOVA, frequently suffer from severe Type I error inflation when the independence assumption is violated by spatial dependence. This paper introduces a novel rank-based test framework that utilizes spatial kernel smoothing to construct robust empirical distribution functions (EDFs). We establish the asymptotic properties of the test statistic under $α$-mixing conditions, proving its convergence to a weighted sum of chi-squared random variables. To facilitate practical inference, we employ a Satterthwaite approximation to derive effective degrees of freedom that account for the spatial 'inflation' of variance. The theoretical framework is developed in detail, providing a rigorous foundation for the proposed method. Simulation studies and applications to real yield quality data are left to future work.

We study the problem of matching correlated VAR time series databases, where a multivariate time series is observed along with a perturbed and permuted version, and the goal is to recover the unknown matching between them. To model this, we introduce a probabilistic framework in which two time series $(x_t)_{t\in[T]},(x^\#_t)_{t\in[T]}$ are jointly generated, such that $x^\#_t=x_{π^*(t)}+σ\tilde{x}_{π^*(t)}$, where $(x_t)_{t\in[T]},(\tilde{x}_t)_{t\in[T]}$ are independent and identically distributed vector autoregressive (VAR) time series of order $1$ with Gaussian increments, for a hidden $π^*$. The objective is to recover $π^*$, from the observation of $(x_t)_{t\in[T]},(x^\#_t)_{t\in[T]}$. This generalizes the classical problem of matching independent point clouds to the time series setting. We derive the maximum likelihood estimator (MLE), leading to a quadratic optimization over permutations, and theoretically analyze an estimator based on linear assignment. For the latter approach, we establish recovery guarantees, identifying thresholds for $σ$ that allow for perfect or partial recovery. Additionally, we propose solving the MLE by considering convex relaxations of the set of permutation matrices (e.g., over the Birkhoff polytope). This allows for efficient estimation of $π^*$ and the VAR parameters via alternating minimization. Empirically, we find that linear assignment often matches or outperforms MLE relaxation based approaches.

Forecasting PM$_{2.5}$ concentration is important to solving air pollution problems in Wuhan. This paper proposes a PM$_{2.5}$ concentration forecast model based on nonlinear regression, including a single-value forecast model and an interval forecast model. The single-value forecast model can precisely forecast PM$_{2.5}$ concentration for the next day, with forecast bias about 6 $μg/m^3$ in goodness of fit analysis. The interval forecast model can efficiently forecast high-concentration and low-concentration days, which covers 60%-80% observed samples in model validation. Moreover, this paper combines the PM$_{2.5}$ concentration forecast model with NCEP Climate Forecast System Version 2 to realize its forecast application, then develops NCEP CFS2's PM$_{2.5}$ concentration forecast model to enhance forecast accuracy. The results indicate that the PM$_{2.5}$ concentration forecast model has good capacity for independent forecasting.

State-space models (SSMs) are the standard formalism for Bayesian treatment of dynamical systems, with natural applications in statistics, signal processing, and machine learning. Despite their importance in both theory and application, dynamical systems have proven difficult to incorporate in modern probabilistic programming languages (PPLs), making state-of-the-art methods less accessible to practitioners and introducing friction in following the "Bayesian workflow." We introduce dynestyx, a probabilistic programming library with first-class support for SSMs, including state-of-the-art methods in the estimation of both states and parameters. Through a single, unified interface, users may specify arbitrary priors for discrete-time or continuous-time dynamical systems, perform inference over mixed-effect data, and make state and parameter estimates with principled uncertainty quantification.

Recovering dynamical systems from noisy observations is a recurring challenge across scientific domains, including neuroscience and physics. Latent stochastic differential equations (SDEs) address this by modeling the system as an unobserved state that evolves according to a learnable SDE and generates the observations. Variational inference (VI) provides a tractable objective for fitting latent SDEs. Traditional VI algorithms evaluate this objective by numerical simulation over a time discretization, trading fidelity for computational cost. A recent class of algorithms, simulation-free VI, sidesteps this tradeoff by parameterizing the posterior through its instantaneous marginals rather than its drift. In this work, we show that the efficiency of existing simulation-free VI algorithms comes at a price: their parameterizations restrict the approximate posterior to a subset of the SDEs available to simulation-based methods, degrading posterior inference and parameter learning. We propose Helmholtz-SDE, a simulation-free VI algorithm that closes this gap by optimizing over path laws compatible with a prescribed collection of marginals. Helmholtz-SDE recovers dynamics more faithfully than prior simulation-free methods, with the largest gains under high posterior uncertainty. It further matches the performance of simulation-based VI at a fraction of the runtime.

Sampling from complex, unnormalized probability densities is a fundamental challenge in Bayesian inference and probabilistic modeling. While Markov chain Monte Carlo (MCMC) methods provide asymptotic guarantees, they often suffer from slow mixing and high computational costs due to fixed or manually tuned trajectory lengths. In this work, we propose a novel framework that treats trajectory termination as a learnable component of the sampling dynamics. By framing MCMC within the theory of non-acyclic generative flow networks (GFlowNets), we train state-dependent neural classifiers to decide when a trajectory has reached a high-density region and should terminate. We theoretically establish the connection between optimal classifiers and the target density via detailed balance conditions and introduce a multilevel training scheme to facilitate exploration in complex geometries. Experimental results across various benchmark densities demonstrate that our approach significantly reduces average trajectory lengths while improving mode coverage and mixing compared to standard MCMC baselines.

Finding D-optimal designs for generalized linear models (GLMs) is challenging due to the dependence of the Fisher information matrix on unknown parameters and the lack of closed-form solutions, particularly when input factors include both discrete and continuous variables. Although classical algorithms and recent metaheuristic approaches have offered partial solutions, there remains a need for robust and computationally efficient methods. In this paper, we propose a penalized Particle Swarm Optimization (PSO) approach, named $p$-PSO. Here we introduce a new, general-purpose penalty formulation for constrained optimization and demonstrate its effectiveness in optimal design problems. The formulation is algorithm-agnostic and applicable to a broad class of black-box optimization methods. Results show that the method is highly efficient, with its primary contribution being a penalty formulation that enables the direct use of an off-the-shelf PSO algorithm and extends naturally to more general constrained optimization tasks.

Bayesian inference for inverse problems is run to evaluate integrals -- posterior expectations, tail probabilities, and risks -- across a stream of observations. The standard estimate averages the integrand over posterior samples, a Monte-Carlo average whose error decays only as the square root of the sample size, so accuracy demands many samples -- prohibitive when each one calls a partial-differential-equation forward model. Mean-shift interacting particles need far fewer: they return a small set of signed-weight nodes -- a deterministic quadrature whose weighted averages estimate those integrals. Finding the nodes, however, is a per-observation optimization that, in its most accurate form, reads the posterior score at every step -- returning the cost it meant to save. We introduce amortized mean-shift interacting particles, a learned map that emits the weighted nodes from an observation and a few posterior samples in a single forward pass. Training asks only for joint parameter-observation samples and a posterior to draw from -- a conditional normalizing flow, an empirical conditional, or any reference the user can sample -- and the map learns to integrate that posterior from samples alone, evaluating neither its density nor its score. Once trained, it generalizes to unseen observations and integrands at any node budget and improves on independent samples in two ways: by reweighting them, provably no worse than the equal weights of Monte-Carlo; and by moving them, which empirically lowers it further. Across closed-form, sampled, learned, and physics-based posteriors -- up to a thousand-coefficient groundwater field -- it integrates more accurately than the same number of samples at every budget, and a posterior-whitened, dimension-aware kernel removes the high-dimensional wall. The result is a Pareto improvement on Monte-Carlo integration, not a competitor to drawing more samples.

This paper explores policy gradient algorithms for training stochastic policies to sample from structured discrete probability distributions under the Generative Flow Network (GFlowNet) framework. Building on extensive theoretical connections between GFlowNets and entropy-regularized reinforcement learning, we derive equivalents of standard policy gradient algorithms for training GFlowNets, as well as experimentally explore their various methodological aspects, including baseline training and advantage estimation. Most importantly, our work is the first to derive and successfully apply proximal policy optimization to GFlowNets, showing its improved convergence speed and data efficiency compared to standard GFlowNet training objectives on benchmarks ranging from synthetic energies to molecular graph generation.

In this paper we investigate the efficacy of the score-based martingale posteriors (SMP) (Cui & Walker, 2025; Fong et al., 2023) in the context of modern and large-scale machine learning problems and its potential for meaningful uncertainty quantification. SMPs work with a stochastic gradient ascent-type recursion on the parameter space of stochastic models and construct a martingale on the parameter space. Under simple mathematical assumptions, the recursion can be built so that the parameters form a martingale sequence which possesses a limiting, in time, random variable, the latter of which can be simulated very quickly, in contrast to Monte Carlo-based methods such as Markov chain Monte Carlo. In this expository paper we explore the SMP for inferring the parameters of deep neural networks (DNNs) and, where feasible, compare our results to the state-of-the-art Monte Carlo methods aimed at inferring conventional Bayesian posteriors.

Stochastic trace estimation is a standard tool for approximating the trace of a large-scale matrix available only through matrix-vector products. However, in tensor-structured settings, unstructured Gaussian or Rademacher test vectors may be prohibitively expensive to store and compute with, while cheaper rank-one tensor-product vectors can require sample complexities that grow exponentially with the tensor order. This work studies Gaussian random tensor train vectors as a structured alternative for stochastic trace estimation. We show that, with a suitable choice of the tensor train rank, random tensor train vectors recover dimension-independent guarantees for the Girard--Hutchinson estimator. In particular, a median-of-means variant with tensor train rank $r \geq d-1$ achieves the same dependence on the accuracy $\varepsilon$ and failure probability $δ$ as the classical estimator based on unstructured Gaussian vectors. We further prove an oblivious subspace injection result for sketches formed from independent Gaussian random tensor train vectors: tensor train rank $r\geq d-1$ and $\mathcal{O}(\varepsilon^{-2}(k+\log(1/δ)))$ samples suffice for a $k$-dimensional target subspace. Finally, we investigate the use of such sketches within the Nyström++ framework. We show that the resulting estimator can achieve the desired $\mathcal{O}(\varepsilon^{-1})$ sample complexity under an additional spectral-tail condition. These results provide clarififcation on both the potential and the limitations of random tensor train vectors in stochastic trace estimation.

Variational inference (VI) is a core engine of modern AI, enabling scalable approximate Bayesian learning and uncertainty-aware training of large probabilistic and generative models. In this paper, we propose Structured Nonparametric Variational Inference (SN-VI), a novel framework for modeling complex dependencies among latent variables in posterior approximation, leveraging multivariate spline techniques. Unlike traditional methods that rely on the mean-field assumption, SN-VI preserves intricate latent variable dependencies, providing a flexible and accurate approximation of posteriors with arbitrary shapes. We establish rigorous theoretical guarantees, including the derivation of the lower bound for the variational objective and proof of asymptotic consistency in posterior estimation. To facilitate practical implementation, we develop an algorithm that automatically identifies dependent latent variables and their underlying dependence structure, without requiring manual specification. Simulation studies validate the effectiveness of SN-VI in approximating posterior distributions with bounded support and complex dependencies. The proposed method has been successfully applied to high-dimensional structured data, including computer vision datasets and spatial transcriptomics. In these applications, SN-VI demonstrates improved generative model performance and effectively uncovers coupled biological signals through the learned dependency structure.

We design a new unconstrained coordinate system where a $p\times p$ symmetric positive definite (SPD) matrix $Θ$ is represented by a reverse telescoping map $Θ(x)=\rm{RT}(x)$, with $x=(v,d,r)\in\mathbb{R}\times\mathbb{R}^{(p-1)}\times\mathbb{R}^{p(p-1)/2}$, representing respectively the log volume or log determinant; and the shape, as encoded by log relative diagonal scales and partial covariances among the nodes. This construction results in important properties not available in other charts, e.g., matrix logarithm, such as Jacobian depending on only the log-determinant. A useful feature of our construction is $x$ contains a lossless symbolic representation of both the matrix and its inverse. Many important computations involving a matrix and its inverse can be performed in $O(p^2)$ in the transformed domain, while it is the rendering of results in matrix forms (on demand) that must incur an $O(p^3)$ cost. Moreover, two unit-determinant matrices in the transformed domain can be joined by a straight line with pathwise unit determinant. For generative modeling, this allows designing a split volume-shape flow model trained by conditional flow matching for transporting the shape over the unit-determinant path, with a separate one-dimensional flow for transporting the volume or the determinant. The forbidding SPD constraint, tamed thus into a powerful guiding force, leads to the surprising insight that it is in some sense easier to design a volume-normalized shape flow for SPD compared to the unconstrained $\mathbb{R}^{p\times p}$, with no intrinsic notion of volume to aid normalization, unlike the determinant of SPD matrices. We apply our construction for up to $p=200$ in generative modeling of SPD matrices on a difficult synthetic bimodal target, and in generating brain connectivity networks by models trained on fMRI data; as well as in intrinsic diffusion on the SPD manifold.

Quadratic Unconstrained Boolean Optimization (QUBO) problems are widespread in both industrial applications and scientific studies. A QUBO problem corresponds to the optimization of a system of Ising spins defined on a generally sparse and heterogeneous graph. When the QUBO problem contains conflicting requests, the corresponding Ising system is frustrated, generating a complex energy landscape, which is hard to explore and optimize. Despite extensive algorithmic and hardware developments, finding low-energy configurations in these systems remains challenging (e.g., local-update heuristics typically become trapped in metastable states), especially when the (possibly frustrated) interactions generate extended correlated domains. We introduce CluMP (Cluster-based Message-Passing), an algorithm that performs collective updates on connected clusters of spins using information from Belief Propagation (BP). By controlling the amount of frustration within clusters, CluMP enables BP convergence on large subgraphs and proposes nonlocal rearrangements involving up to hundreds of spins in a single move. We benchmark CluMP against state-of-the-art local-update heuristics on spin-glass models defined on several graph topologies, including random regular graphs and lattice regular graphs in two and three dimensions. Cluster moves consistently bypass local trapping and reach lower energies with fewer effective operations than single-spin dynamics. These results demonstrate that frustration-tolerant cluster updates can be implemented efficiently on sparse graphs. The CluMP framework provides a scalable strategy for large-scale combinatorial optimization and inference problems, where exploiting medium- and long-range correlations is key to navigating complex energy landscapes.

Moment-constrained maximum entropy (MaxEnt) reconstructs probability densities from a few moments in uncertainty evaluation (GUM) and reliability analysis. The classical method uses monomial constraints x^i. We show that monomials are merely one choice of generating element of the underlying Kunchenko decomposition space, and that this choice -- more than the solver -- governs which densities are representable and how well-conditioned the dual problem is. We study three elements under one dual solver: a fractional-power element (PATP) that reduces fractional-moment exponent selection to a one-dimensional scan on signed supports; a trigonometric (characteristic-function) element whose constraints exist for every distribution and keep the dual Hessian bounded; and a logarithmic-rational element log(1+(x/s)^2) whose single constraint yields the Student/Cauchy family (1+(x/s)^2)^lambda, representing algebraic tails the first two do not produce. A parity-admissibility theorem shows that an element of odd functions cannot represent any non-uniform symmetric density; the unifying lesson is a design map matching the element to the target's tail class. Empirically, on a bimodal Gaussian mixture the scan-selected fractional member cuts reconstruction MSE by 8.5x over the six-moment monomial baseline (all 20 seeds), while the trigonometric element is best-conditioned. On heavy tails the fractional element restores feasibility where monomial MaxEnt is infeasible (19/20 seeds) and reconstructs the body (KS 0.068) but not the tail, whereas the matched logarithmic element recovers the Cauchy tail index from one constraint. A variance-optimal rule (oPMM-alpha) selects the element for the reported functional. An analytical product-moment evaluator makes a measurement-and-verification optimization fitness exactly deterministic and faster than Monte Carlo, removing its noise-induced violations.

Efficient event generation is a major computational challenge for precision collider phenomenology, especially for high-multiplicity final states where matrix-element evaluations are expensive and rejection-sampling efficiencies are low. We study an alternative approach based on many parallel underdamped Langevin chains, retaining one terminal state from each chain to obtain unweighted events while avoiding within-chain autocorrelation. A learned Stein discrepancy is used as a convergence diagnostic, providing a data-driven estimate of the relaxation time. We apply the method to tree-level $u\bar u\to Z+n g$ event generation and find that relaxation requires only a modest number of exact-target Langevin steps, with mild growth over the multiplicities studied. Finally, we show that simple neural-network surrogate initialization can substantially reduce the required number of exact matrix-element and gradient evaluations.

This paper focuses on accelerating Markov chain Monte Carlo sampling in Bayesian inverse problems in which forward model evaluations dominate the computational cost. It builds on several established ingredients previously used in related scenarios: delayed acceptance, neural network surrogate models, Hamiltonian proposals, and proposal subchains. The main framework is the delayed-acceptance Metropolis-Hastings algorithm of Christen and Fox (2005). The first-stage proposal distribution is constructed from a subchain of Hamiltonian trajectories targeting the surrogate posterior. For each fixed surrogate model, the Hamiltonian subchain and delayed-acceptance correction define a kernel invariant with respect to the exact posterior. In the present work, the surrogate is updated only during a burn-in phase, after which the production run uses a fixed surrogate model. The sampling framework is implemented in Python using parallel processes. Several chains are generated in parallel and share a single surrogate model trained during burn-in on all collected data. The forward model is treated as a black box; therefore, the application area is broad. However, the main motivation is efficient solution of geotechnical inverse problems with material properties represented by Gaussian random fields. In this study, the sampling framework is applied to a geotechnical inverse problem in which hydraulic conductivity and porosity are modeled as non-stationary Gaussian random fields approximated using truncated Karhunen-Loeve expansions. Based on a precomputation, the truncation dimensions are chosen separately for hydraulic conductivity and porosity. The forward model outputs are pore pressure values at control points and selected observation times. These are compared with in situ pore pressure measurements collected over one year during the Tunnel Sealing Experiment in an underground laboratory in Canada.

Diffusion Monte Carlo (DMC) and Monte Carlo for particle transport with importance sampling both involve simulations of weighted walkers that undergo birth and death processes (splitting and Russian Roulette). The established implementations of these methods are quite different: Particle simulation Monte Carlo employs a stack to handle the splitting history whereas in traditional DMC one follows a swarm of walkers. The particle simulation Monte Carlo approach involves a depth first traversal of the visited configurations whereas the traditional DMC approach may be seen as a breadth first traversal. In the present work the implementation of a depth first, stack based approach to DMC is described and a complete code is presented. The depth first approach, called DMCD here, can be more memory efficient than the breadth first approach, both for total memory and for use of a memory hierarchy and of co-processors. The implementation appears very natural for population control and for descendant weighting and it unifies algorithmic treatment of the eigenvalue problem (DMC) with the linear equation problem (particle transport). A concern with DMCD that is not present in the breadth first approach, and that is successfully addressed here, is the need to maintain a pool of starters for use when a new walker is required and the stack is empty. The DMCD approach appears to have the potential to become the preferred implementation for many DMC applications.

Quantum machine learning increasingly relies on pure-state representations, motivating generative models that sample directly in quantum representation space rather than perturbing classical inputs and re-encoding. We introduce Stochastic Schrödinger Diffusion Models (SSDMs), a score-based generative framework that defines diffusion, scores, and reverse-time sampling intrinsically on the complex projective manifold $\mathbb{CP}^{d-1}$ under the Fubini--Study metric. SSDMs combine a Riemannian Ornstein--Uhlenbeck forward diffusion with a stochastic Schrödinger realization, and learn reverse-time dynamics driven by the Riemannian score. Our central technical contribution is a local-time learning objective that exploits the local Euclidean OU limit of intrinsic manifold diffusions in Fubini-Study normal coordinates to obtain an analytic teacher score, bypassing the intractable transition densities that limit existing Riemannian score-based models. Across synthetic, physics-inspired (TFIM, XXZ), and quantum feature-state benchmarks up to $14$ qubits, SSDMs match target pure-state ensembles by orders of magnitude on MMD and observable statistics over both ambient Euclidean and matched Riemannian score-based baselines, and improve representation-level diagnostics for downstream quantum kernel methods.

Stochastic reduced-order models are widely used to represent the effective dynamics of complex systems, but estimating their drift and diffusion coefficients from data remains challenging. Standard approaches often rely on short-time trajectory increments, state-space partitioning, or repeated simulation of candidate models, which become unreliable or computationally expensive for high-dimensional systems, coarse temporal sampling, or unevenly sampled data. We introduce a data-driven calibration method based on a novel relationship between the coefficients of a stochastic reduced model and the conditional score of the finite-time transition density, defined as the gradient of the logarithm of the transition density with respect to the initial state. The resulting identity expresses derivatives of lagged correlation functions as stationary expectations over observed lagged pairs involving this conditional score and the unknown model coefficients. This formulation allows the drift and diffusion structure to be constrained directly from finite-lag statistics, without differentiating trajectories, partitioning state space, or repeatedly integrating candidate reduced models during calibration, yielding a least-squares fitting problem over stationary lagged pairs. We validate the approach on three systems of increasing complexity: an analytically tractable Cox--Ingersoll--Ross diffusion, a two-dimensional nonequilibrium diffusion with affine multiplicative noise, and a periodic soft-spin stochastic Landau--Lifshitz chain. Across these tests, the inferred models preserve the invariant statistics while reproducing finite-lag dynamical correlations. The framework provides a scalable route for learning stochastic reduced-order models from data that reproduce prescribed statistical and dynamical properties.

Nonlinear stochastic motion presents significant challenges for Bayesian particle tracking. To address this challenge, we propose a lifting framework that constructs a higher-dimensional linear stochastic representation of nonlinear state-space models. The resulting surrogates enable the use of computationally efficient linear filtering techniques while retaining a direct connection to the underlying nonlinear dynamics. The paper derives the necessary conditions for such transformations using Ito's lemma and variational calculus, and illustrates the method on a bistable cubic motion model, radial Brownian process model, and a logistic model with multiplicative noise. Simulations confirm that the transformed linear systems, when projected back, accurately reconstruct the nonlinear dynamics and, in distinct regimes of stiffness and singularity, yield tracking accuracy competitive with conventional filters, while avoiding their structural instabilities.

Copula models are widely employed in multivariate time series analysis because they permit flexible modelling of marginal distributions independently of the dependence structure, which is fully characterised by the copula function. However, Bayesian inference with these models becomes computationally demanding as the number of variables in the time series increases. Motivated by the classical inference functions for margins (IFM) approach, we propose a new neural-network based inference framework for estimating parameters in copula models, termed the neural inference functions for margins (N-IFM). N-IFM enables rapid parameter estimation for new data, fast sequential prediction, and efficient model comparison via time-series validation. We assess the performance of N-IFM using both simulated and real datasets and compare it to Hamiltonian Monte Carlo, demonstrating substantial computational gains with comparable inferential accuracy.

The performance of the Monte Carlo sampling methods relies on the crucial choice of a proposal density. The notion of optimality is fundamental to design suitable adaptive procedures of the proposal density within Monte Carlo schemes. This work is an exhaustive review around the concept of optimality in importance sampling. Several frameworks are described and analyzed, such as the marginal likelihood approximation for model selection, the use of multiple proposal densities, a sequence of tempered posteriors, and noisy scenarios including the applications to approximate Bayesian computation (ABC) and reinforcement learning, to name a few. Some theoretical and empirical comparisons are also provided.

We propose a new gradient-free method for infinite-dimensional optimization in Hilbert spaces that requires only the computation of directional derivatives. Though functional optimization is often solved through finite-dimensional gradient descent over a parametrization, such as neural networks, we instead propose to leverage the functional nature of the optimization problem to enable provable guarantees. However, infinite-dimensional gradients are often hard to compute in practice, rendering naïve functional gradient descent intractable. To overcome this limitation, our framework leverages only directional derivatives and a pre-basis for the Hilbert space, i.e., a linearly independent set whose span is dense. This resolves the tractability issue, as pre-bases are much more accessible than full orthonormal bases or reproducing kernels -- which may not even exist -- and individual directional derivatives can be computed using automatic differentiation. We showcase the use of our method to solve partial differential equations à la physics-informed neural networks (PINNs), where it effectively enables provable convergence.

Computer simulations are indispensable for analyzing complex systems, yet high-fidelity models often incur prohibitive computational costs. Multi-fidelity frameworks address this challenge by combining inexpensive low-fidelity simulations with costly high-fidelity simulations to improve both accuracy and efficiency. However, certain scientific problems demand even more accurate results than the highest-fidelity simulations available, particularly when a tuning parameter controlling simulation accuracy is available, but the exact solution corresponding to a zero-valued parameter remains out of reach. In this paper, we introduce the Diffusion Non-Additive (DNA) model, inspired by generative diffusion models, which captures nonlinear dependencies across fidelity levels using Gaussian process priors and extrapolates to the exact solution. The DNA model: (i) accommodates complex, non-additive relationships across fidelity levels; (ii) employs a nonseparable covariance kernel to model interactions between the tuning parameter and input variables, improving predictive performance; (iii) provides closed-form expressions for the posterior predictive mean and variance, allowing efficient inference and uncertainty quantification; and (iv) establishes rigorous theoretical bounds on the prediction error, leading to an optimal experimental design strategy. The methodology is validated on a suite of numerical studies and real-world case studies. An R package implementing the proposed methodology is available to support practical applications.

We introduce Prob-GParareal, a probabilistic extension of the GParareal algorithm designed to provide uncertainty quantification for the Parallel-in-Time (PinT) solution of (ordinary and partial) differential equations (ODEs, PDEs). The method employs Gaussian processes (GPs) to model the Parareal correction function, in line with GParareal, further enabling the propagation of numerical uncertainty across time and yielding probabilistic forecasts of the system's evolution. Furthermore, Prob-GParareal accommodates probabilistic initial conditions and maintains compatibility with classical numerical solvers, ensuring its straightforward integration into existing Parareal frameworks. Here, we first conduct a theoretical analysis of the computational complexity and derive error bounds of Prob-GParareal. Then, we numerically demonstrate the accuracy and robustness of the proposed algorithm on five benchmark ODE systems, including chaotic, stiff, and bifurcation problems. To showcase the flexibility and potential scalability of the proposed algorithm, we also consider Prob-nnGParareal, a variant obtained by replacing the GPs in Parareal with the nearest-neighbors GPs, illustrating its increased performance on an additional PDE example. This work bridges a critical gap in the development of probabilistic counterparts to established PinT methods.

Diffusion and flow-based models learn powerful data priors by training a denoiser to reverse Gaussian corruption. To use this prior to solve a linear inverse problem, one needs to sample from the posterior, but the score that the prior provides is the unconditional score, not the posterior score. Existing methods either steer a fixed pretrained denoiser with approximate measurement-matching corrections, or train a conditional restoration model that abandons the denoising structure of the prior. We derive the exact posterior score in closed form for linear Gaussian inverse problems under general Gaussian interpolants, and show that posterior sampling reduces to a denoising problem at an operator-dependent shifted pivot under an anisotropic noise covariance. We turn this identity into Exact Posterior Score (EPS), a denoising training objective that preserves the input/output structure of standard pretraining and can therefore be trained from scratch or fine-tuned from a pretrained denoiser. At inference, EPS uses the same sampler as the underlying backbone, with no likelihood gradients or projections. We evaluate EPS on five linear inverse problems across FFHQ and ImageNet, where it outperforms training-free and training-based baselines on fidelity, perceptual, and distributional metrics, while using roughly an order of magnitude fewer denoiser evaluations than gradient-based posterior samplers.

In this work, we investigate new activation functions for achieving arbitrary-accuracy Sobolev approximation by fixed-size neural networks. We first show that any function in $W^{2,\infty}((a,b)^d)$ can be approximated with arbitrary accuracy, measured in the $W^{1,\infty}$-norm, by a fixed-size neural network using the Elementary Universal Activation Function ($\mathrm{EUAF}$). To extend this result to $W^{s,\infty}((a,b)^d)$ for $s\in\mathbb{N}$, we introduce a smooth activation $\mathrm{DUAF}_{\infty}$ from the family of Differentiable Universal Activation Functions ($\mathrm{DUAF}_n$). We prove that any function in $W^{s,\infty}((a,b)^d)$ can be approximated with arbitrary accuracy in the $W^{s-1,\infty}$-norm by a fixed-size $\mathrm{DUAF}_{\infty}$-activated network. We further construct sigmoidal variants $\widetilde{\mathrm{DUAF}}_n$ and show that, for every $1\leq s\leq n$, fixed-size $\widetilde{\mathrm{DUAF}}_n$-activated networks still approximate any $f\in W^{s,\infty}((a,b)^d)$ with arbitrary accuracy in the $W^{s-1,\infty}$-norm. In all these results, the width and depth bounds are computed explicitly, and the proposed activations are elementary.

Functional optimization problems are typically solved by optimizing the parameters of a fixed representation, such as a neural network, resulting in highly nonconvex losses that complicate both training and theoretical analysis. An interesting alternative is functional gradient descent (FGD), that is, gradient descent directly in function space, which benefits from strong convergence results and admits a clean theory. However, FGD is difficult to implement in practice because functional gradients are infinite-dimensional, and thus cannot be fully computed nor stored in memory. Existing implementations therefore rely on fixed approximations, which introduce approximation error. We propose a new, theoretically-grounded FGD algorithm that adapts the representation of the functional gradients over the course of optimization. By explicitly incorporating this approximation into the analysis, we establish convergence to a stationary point (for smooth losses) and to a global minimizer (under smoothness + a Polyak-Lojasiewicz-type condition) regardless of our approximations. To the best of our knowledge, this is the first implementable FGD method with such guarantees in a general setting. We demonstrate the effectiveness of our method on regression, numerical solution of PDEs, and modern computer vision. Across settings, our method consistently outperforms both FGD with fixed approximations and neural network baselines in efficiency and accuracy.

Causal self-attention is a coupling mechanism: each token's hidden state is updated by a learned mixture of preceding tokens at the same timescale. This paper asks whether a second, temporally slower coupling-a slow sub-system operating on a temporally-downsampled view of the sequence and fed back into the fast path through a zero-initialised gate-complements it. The question is framed in the language of singularly perturbed ordinary differential equations (ODEs), where the fast variable $x$ evolves at the token rate, the slow variable $y$ evolves at one update per $P$ tokens, and the timescale ratio $\varepsilon = 1/P$ is enforced structurally by causal block-mean pooling. The paper instantiates the fast-slow ODE formalism as a concrete neural network: a fast path of standard causal attention over $T$ tokens, a slow path of full attention over $T/P$ pooled tokens ($P^2 \times$ cheaper per layer), and a zero-initialised additive gate. In addition, under a linear-generator assumption on the fast dynamics, we prove that the equilibrium manifold $x = ϕ(y)$ is exactly the master-equation (ME) stationary distribution $p_{\mathrm{st}}(y)$; in that regime a learned MLP $ϕ_θ(y)$ is a variational approximation of it (the trained block is not a generator, so this identity is the structured limit, not a claim about the network as trained). Empirically, at $500$k tokens the coupling is neutral -- the gate stays closed and the coupled and frozen ablations are within run-to-run noise -- at a wall-clock cost comparable to a dense baseline. The contribution is the precise, gap-marked mapping itself, not a performance gain.

Diffusion Flow Matching (DFM) has recently emerged as a versatile framework for generative modeling, yet its theoretical convergence properties remain only partially understood. In this work, we provide refined and novel convergence guarantees for Brownian motion based DFMs, focusing on the discretization error. Our analysis is conducted under the Kullback-Leibler (KL) divergence and the 2-Wasserstein distance. Under finite-moment conditions and a mild score integrability assumption, we derive KL convergence bounds with improved dimensional dependence compared to prior work, achieving, up to our knowledge, state-of-the-art scaling under minimal conditions. We further extend the analysis to the 2-Wasserstein distance: under an additional first-order score integrability assumption and a weak log-concavity condition, we obtain convergence guarantees with dimensional dependence consistent with the KL case.

Domain Generalization (DG) aims to train models that generalize to unseen target domains but often overfit to domain-specific features, known as undesired correlations. Gradient-based DG methods typically guide gradients in a dominant direction but often inadvertently reinforce spurious correlations. Recent work has employed dropout to regularize overconfident parameters, but has not explicitly adjusted gradient alignment or ensured balanced parameter updates. We propose GENIE (Generalization-ENhancing Iterative Equalizer), a novel optimizer that leverages the One-Step Generalization Ratio (OSGR) to quantify each parameter's contribution to loss reduction and assess gradient alignment. By dynamically equalizing OSGR via a preconditioning factor, GENIE prevents a small subset of parameters from dominating optimization, thereby promoting domain-invariant feature learning. Theoretically, GENIE balances convergence contribution and gradient alignment among parameters, achieving higher OSGR while retaining SGD's convergence rate. Empirically, it outperforms existing optimizers and enhances performance when integrated with various DG and single-DG methods.

We introduce, to our knowledge, the first deep generative modeling framework for probability distributions continuously supported on compact metric graphs. Given source and target measures on a metric graph, our method embeds the graph into a smooth ambient space, solves an entropic Kantorovich problem via a neural semidual parameterization, and projects generated samples back onto the original graph. We study two embedded geometries: an extrinsic Euclidean realization and the intrinsic tropical Abel--Jacobi embedding into the Jacobian torus. In both cases, the resulting generator is graph-supported by construction. We prove that, in the joint limit of increasing neural expressivity, the learned generator converges weakly to a valid transport coupling between the original graph measures. Empirically, across a range of geometrically distinct graphs, our method matches or improves upon heuristic transport baselines based on discrete graph OT, while scaling more favorably. Finally, we demonstrate scalability on real-world urban mobility data by training our model on one million Uber pickup locations in Manhattan, New York City.

Flow matching is a powerful generative modeling framework, valued for its simplicity and strong empirical performance. However, its standard formulation treats signals on structured spaces, such as fMRI data on brain graphs, as points in Euclidean space, overlooking the rich topological features of their domains. To address this, we introduce topological flow matching, a topology-aware generalization of flow matching. We interpret flow matching as a framework for solving a degenerate Schrödinger bridge problem and inject topological information by augmenting the reference process with a Laplacian-derived drift. This principled modification captures the structure of the underlying domain while preserving the desirable properties of flow matching: a stable, simulation-free objective and deterministic sample paths. As a result, our framework serves as a drop-in replacement for standard flow matching. We demonstrate its effectiveness on diverse structured datasets, including brain fMRIs, ocean currents, seismic events, and traffic flows.

This paper studies the information gap between mixture detection and label recovery in binomial logistic mixtures. Standard likelihood-based criteria such as the Bayesian information criterion (BIC) can detect the presence of two components, but this does not guarantee that the corresponding labels are recoverable. We show that this gap is intrinsic to binomial logistic mixtures with a fixed number of trials: observed-data evidence for mixture structure and per-observation information for label recovery have different local orders in the component separation, and only the former accumulates with the sample size. As a result, there exists a detectable-but-unrecoverable regime in which BIC selects two components while the posterior labels remain essentially uninformative. To address this issue, we propose two feasibility-aware inference procedures: a recoverability-aware BIC with a posterior-entropy penalty and an entropy-regularized estimator that mitigates the tendency of the maximum likelihood estimator to produce overly separated components and overly concentrated posterior responsibilities. Numerical experiments confirm the predicted gap and demonstrate that the proposed methods avoid misleading component selections and improve the calibration of posterior label probabilities.

Test-time training (TTT) adapts a pretrained model to each prompt via parameter updates, improving accuracy under pretraining-to-test distribution shifts. Yet, its performance often suffers from instability and sensitivity to hyperparameters such as update steps and subspace. We explain this behavior through a decision-theoretic lens, treating TTT as implicit Bayesian inference in the kernel regime. Under a Gaussian process benchmark, we show that TTT reduces prediction error when updates are spectrally matched to the prompt's signal-to-noise ratio and aligned with query-relevant eigen-directions. This perspective underpins the following results: (1) we show when fixed update steps and subspaces fail under distribution shifts, motivating adaptive strategies; (2) we prove that selecting update steps via prompt evidence admits a PAC-Bayes guarantee against overfitting; and (3) we characterize the Bayes-optimal update subspace under a linear-Gaussian correction model, yielding a scoring rule for selecting Transformer blocks and heads. Our theory helps explain the empirical instability of TTT, taking a step toward principled guidance for when, how far, and which directions to adapt.

In large language model (LLM) serving, each request accumulates persistent graphics processing unit (GPU) memory during service as its key-value cache grows with every generated token. Under high concurrency, aggregate memory usage therefore increases endogenously over time: the service process itself creates future capacity pressure. When memory capacity is exceeded, systems evict active requests, discarding cached state and restarting them later, which wastes computation and reduces throughput. We develop a discrete-time dynamical model of memory-constrained LLM inference that captures admission, memory growth, and eviction under continuous batching. In the saturated-input regime, the system admits both eviction-free fixed points and limit cycles with evictions. For homogeneous workloads, we show that the eviction-free equilibrium is unstable and that, except for a Lebesgue-measure-zero exact-capture set, the system converges to a unique worst-case limit cycle that is asymptotically stable outside this exceptional set, with throughput losses as large as 50%. For heterogeneous workloads, we prove a stability criterion in the two-class common-input setting and explain how the survival-polynomial mechanism generalizes to multiple classes and heterogeneous-input lengths. Under an input-dominated scaling regime, coprime decoding lengths stabilize the eviction-free equilibrium, while non-coprime lengths create synchronized modes that drive instability. These results characterize when workload heterogeneity desynchronizes completions and helps stabilize memory-constrained serving. More broadly, we identify service-induced congestion as a structural instability mechanism and derive scheduling design principles for sustaining high throughput.

Ricci flow is a curvature-guided diffusion process that deforms space by shrinking regions of high positive curvature and expanding those with negative curvature. Similarly, discrete Ricci flow on weighted graphs modifies edge weights by shrinking edges with positive Ricci curvature and stretching those with negative Ricci curvature, effectively increasing the separation between clusters. Inspired by these two cornerstone works, we propose a geometry-based RAG reranker enhancement procedure called Ricci-Filtration. By modeling the input query and initial retrieved chunks as a network, where the input query and chunks serve as nodes and embedding-based pairwise relations define an initial graph, Ricci-Filtration leverages discrete curvature and Ricci flow to evaluate the structural importance of each chunk with respect to the user query. The system first filters the initial chunks based on their geometric curvature relative to the query; then, a reranker processes the remaining chunks to enhance generative performance. We theoretically prove that normalized discrete Ricci flow can detect community structures by identifying distinct asymptotic behaviors in edge weights. This supports the removal of ``noisy'' document chunks characterized by large weights and negative Ricci curvature relative to the query node. Extensive experiments confirm that Ricci-Filtration outperforms several baseline reranking methods in accuracy, precision, recall, and F1 scores. Furthermore, ablation studies demonstrate that the Ricci-Filtration generally outperforms the baseline under various settings, highlighting the framework's robustness across different architectures.

In this work, we tackle the following question: Can neural networks trained with gradient-based methods achieve the optimal computational-statistical tradeoff in learning Gaussian single-index models? Prior research has shown that any polynomial-time algorithm under the statistical query (SQ) framework requires $Ω(d^{s^\star/2}\lor d)$ samples, where $s^\star$ is the generative exponent representing the intrinsic difficulty of learning the underlying model. However, it remains unknown whether neural networks can achieve this sample complexity. Inspired by prior techniques such as label transformation and landscape smoothing for learning single-index models, we propose a unified gradient-based algorithm for training a two-layer neural network in polynomial time. Our method is adaptable to a variety of loss and activation functions, covering a broad class of existing approaches. We show that our algorithm learns a feature representation that strongly aligns with the unknown signal $θ^\star$, with sample complexity $\widetilde{O} (d^{s^\star/2} \lor d)$, matching the SQ lower bound up to a polylogarithmic factor for all generative exponents $s^\star\geq 1$. Furthermore, we extend our approach to the setting where $θ^\star$ is $k$-sparse for $k = o(\sqrt{d})$ by introducing a novel weight perturbation technique that leverages the sparsity structure. We derive a corresponding SQ lower bound of order $\widetildeΩ(k^{s^\star})$, matched by our method up to a polylogarithmic factor. Our framework, especially the weight perturbation technique, is of independent interest, and suggests potential gradient-based solutions to other problems such as sparse tensor PCA.

Offline model-based optimization (MBO) proposes candidates by optimizing a surrogate trained on a fixed historical dataset. Because candidates are deliberately out-of-distribution, surrogate rankings are least reliable exactly where the optimizer is most aggressive, yet existing methods provide no per-candidate statistical certificate that a design meets a target threshold. We propose \emph{Conformal Candidate Certification} (CCC), a post-hoc wrapper that attaches a calibrated one-sided lower bound to each candidate and advances only those whose bound exceeds the target. We show that entropy-regularized surrogate maximization induces a Gibbs-tilted proposal, so the same surrogate supplies importance weights for weighted conformal prediction without a separate density-ratio estimation step. In a controlled synthetic study, CCC certifies $16.7\%$ of an aggressive proposal pool with empirical coverage 0.990 at nominal 0.90, while standard conformal prediction ignoring the covariate shift collapses to 0.416 coverage.

Modern recommendation systems increasingly rely on dynamically routing diverse queries to multiple embedding models. Despite its practical significance, this problem remains poorly understood under realistic conditions like adversarial queries, bandit feedback, and limited observability of models. We formalize embedding model routing as an adversarial contextual linear bandit with low-rank experts, where contexts are queries, actions are items, and experts are the embedding models working on low-rank latent representation spaces. We first establish that standard regret notions suffer from structural misspecification or statistical intractability, and we identify a log-quadratic policy class that is expressive enough to capture query-dependent model routing, yet structured enough to allow efficient online learning. Second, we propose a policy gradient algorithm called Hypentropy Policy Gradient (HPG). It provably adapts to the unknown low-rank structure under incomplete information and attains $\tilde{\mathcal O}(s\sqrt{M T})$ linearized policy regret -- where $s, M$, and $T$ are the intrinsic rank of the experts, the number of models, and the number of rounds -- thus avoiding a curse of dimensionality. Finally, we also provide an computationally efficient and parameter-free implementation of HPG.

Molecular dynamics (MD) simulations generate trajectories in a high-dimensional configuration space whose analysis critically depends on molecular descriptors, typically handcrafted observables or learned kinetic embeddings. Designing descriptors that are both expressive and broadly applicable, however, remains challenging. We study persistent homology (PH) as a general-purpose representation for MD and introduce the masked Flood complex, a protein-tailored modification of a recently introduced simplicial complex construction that emphasizes inter-residue structure at low computational cost. Vectorized persistence diagrams then provide information-rich, geometry-aware summaries of protein conformations, which we evaluate on protein class prediction, frame-level observable regression, and Markov state model (MSM) estimation from learned low-dimensional coordinates in a single shared representation space. Results on the mdCATH dataset show that PH-based descriptors are competitive across tasks, with masked Flood PH yielding the most consistent overall performance. Further, when using topologically-informed MSMs as a drop-in replacement within the recent MarS-FM framework for generative modeling of protein conformations, we obtain consistently better ensemble statistics than MSMs based on physical observables. Finally, we explore the transferability of the generative model to qualitatively different, fast folding, proteins.

We study the sample complexity of learning in average-reward weakly-coupled Markov decision processes (WCMDPs) and Restless Bandits (RBs) under a generative model. Naive reduction to a tabular MDP leads to high complexity bounds as the state-action space is exponentially large in the number of arms $N$. By exploiting the weakly coupled structure, we show that near-optimal policies can be learned with sample and computational complexities that are polynomial in $N$. Specifically, we analyze the plug-in approach, which applies an efficient planning algorithm to an empirical model estimated from data. For fully heterogeneous WCMDPs, we establish the first finite-sample PAC guarantee with polynomial complexity and an $O(1/\sqrt{N})$ optimality gap. For homogeneous RBs, we further prove that a smaller optimality gap is achievable under mild structural assumptions. A primary technical contribution of our work is a novel Lyapunov-based analysis framework. Unlike classical approaches that rely on the difficult-to-control bias function, our framework uses an explicitly constructed Lyapunov function along with a drift transfer technique between the true and empirical models. A key step of independent interest in our framework is a fine-grained perturbation analysis for the underlying linear programming (LP) relaxation, which provides a general tool for analyzing LP-based policies and weakly-coupled systems.

In the era of Explainable Artificial Intelligence, there is a renewed focus on single trees for their ease of interpretation. This paper introduces Simultaneous Latent Budget Trees, a probabilistic machine learning framework for classification trees in the presence of a stratification factor such as a temporal, spatial, or demographic variable, acting as a control variable or potential confounder. Standard tree growth procedures are not designed to optimize a conditional split rule. A model-based split rule is proposed in which child nodes are interpreted as latent components of a simultaneous mixture model, such as the Simultaneous Latent Budget Model and its constrained versions, fitted to the parent node. Mixing parameters drive the observations, differently for each group, to the child nodes whereas latent budgets parameters update the response classes profile of each level of the control variable. Parameters are estimated by least squares considering a neural network perspective of the model. An informative tree structure can be interactively visualized with interpretation aids on the node and the paths, including visual pruning and decision tree selection procedure. Suitable measures are proposed to handle an unbalanced response class distribution. The proposed methodology is applied to investigate gender-related differences in disease progression of Amyotrophic Lateral Sclerosis. The SLBT library with the various tree-based algorithms is available in the linked GitHub repository.

Detecting observations from a minority class under severe class imbalance is a central challenge in applications such as fraud detection, medical screening, and industrial quality control. In these settings, each positive prediction triggers a costly follow-up action, an MRI scan, a transaction audit, whose execution is subject to real operational constraints. This paper proposes a formal classification framework under capacity constraints: given a user-defined bound limit $b$ on the proportion of observations that can be labeled as belonging to the minority class, the goal is to find the classifier that maximizes sensitivity on that class. We characterize the optimal classifier under this constraint and establish its equivalence with the classical Bayes classifier under a reweighting of the prior probabilities. We also introduce a capacity-adjusted performance metric $M$ that accounts for the effective detection rate when the capacity constraint is binding. The framework is implemented on top of standard learning methods, k-NN, SVM, random forests, and neural networks, and statistical consistency is established for each. We further show that these methods reduce to post-hoc thresholding when no hyperparameters are oriented toward the capacity-constrained objective, and introduce a capacity-aware support vector machine that exploits the constraint during training and achieves the strongest empirical performance. Experiments on the Taiwanese credit card default dataset confirm that capacity-constrained classifiers substantially outperform both classical approaches and SMOTE under high imbalance regimes. The framework extends naturally to multiclass settings and online environments.

We propose GradPower, a lightweight gradient-transformation technique for accelerating language model pre-training. Given a gradient vector $g=(g_i)_i$, GradPower first applies the elementwise sign-power transformation: $φ_p(g)=({\rm sign}(g_i)|g_i|^p)_{i}$ for a fixed $p>0$, and then feeds the transformed gradient into a base optimizer. Notably, GradPower requires only a single-line code change and no modifications to the base optimizer's internal logic, including the hyperparameters. When applied to Adam (termed AdamPower), GradPower consistently achieves lower terminal loss across diverse architectures (LLaMA, Qwen2MoE), parameter scales (66M to 2B), datasets (C4, OpenWebText), and learning-rate schedules (cosine, warmup-stable-decay). The most pronounced gains are observed when training modern mixture-of-experts models with warmup-stable-decay schedules. GradPower also integrates seamlessly with other state-of-the-art optimizers, such as Muon, yielding further improvements. Finally, we provide theoretical analyses that reveal the underlying mechanism of GradPower and highlight the influence of gradient noise.

Representation Autoencoders (RAE) replace traditional VAE with pretrained vision encoders. In this paper, we systematically investigate several design choices and find three insights which simplify and improve RAE. First, we study a generalized formulation where the representation is defined as sum of the last k encoder layers rather than solely the final layer. This simple change greatly improves reconstruction without encoder finetuning or specialized data (e.g., text, faces). Second, we study the prevalent assumption that RAE (using pretrained representation as encoder) replaces representation alignment (REPA), which distills the same representation to intermediate layers instead. Through large-scale empirical analysis, we uncover a surprising finding: RAE and REPA exhibit complementary working mechanisms, allowing the same representation to be used as both encoder and target for intermediate diffusion layers. Finally, the original RAE struggles with classifier-free guidance (CFG) and requires training a second, weaker diffusion model for AutoGuidance (AG). We show that REPA itself can be viewed as x-prediction in RAE latent space. By simply re-parameterizing the output of the DiT model, it can provide guidance for "free". Overall, RAEv2 leads to more than 10x faster convergence over the original RAE, achieving a state-of-the-art gFID of 1.06 in just 80 epochs on ImageNet-256. On FDr6, RAEv2 achieves a state-of-the-art 2.17 at just 80 epochs compared to the previous best 3.26 (800 epochs) without any post-training. This motivates EPFID@k (epochs to reach unguided gFID < k) as a measure of training efficiency. RAEv2 attains an EPFID@2 of 35 epochs, versus 177 for the original RAE. We also validate our approach across diverse settings for text-to-image generation and navigation world models, showing consistent improvements. The code is available at https://raev2.github.io.

Diffusion and flow matching approaches to generative modeling have shown promise in domains where the state space is continuous, such as image generation or protein folding & design, and discrete, exemplified by diffusion large language models. They offer a natural fit when the number of elements in a state is fixed in advance (e.g. images), but require ad hoc solutions when, for example, the length of a response from a large language model, or the number of amino acids in a protein chain is not known a priori. Here we propose Branching Flows, a generative modeling framework that, like diffusion and flow matching approaches, transports a simple distribution to the data distribution. But in Branching Flows, the elements in the state evolve over a forest of binary trees, branching and dying stochastically with rates that are learned by the model. This allows the model to control, during generation, the number of elements in the sequence. We also show that Branching Flows can compose with any flow matching base process on discrete sets, continuous Euclidean spaces, smooth manifolds, and `multimodal' product spaces that mix these components. We demonstrate this in three domains: small molecule generation (multimodal), antibody sequence generation (discrete), and protein backbone generation (multimodal), and show that Branching Flows is a capable distribution learner with a stable learning objective, and that it enables new capabilities.

Heatmap-based solvers have emerged as a promising paradigm for Combinatorial Optimization (CO). However, we argue that the dominant Supervised Learning (SL) training paradigm suffers from a fundamental objective mismatch: minimizing imitation loss (e.g., cross-entropy) does not guarantee solution cost minimization. We dissect this mismatch into two deficiencies: Decoder-Blindness (being oblivious to the non-differentiable decoding process) and Cost-Blindness (prioritizing structural imitation over solution quality). We empirically demonstrate that these intrinsic flaws impose a hard performance ceiling. To overcome this limitation, we propose CADO (Cost-Aware Diffusion models for Optimization), a streamlined Reinforcement Learning fine-tuning framework that formulates the diffusion denoising process as an MDP to directly optimize the post-decoded solution cost. We introduce Label-Centered Reward, which repurposes ground-truth labels as unbiased baselines rather than imitation targets, and Hybrid Fine-Tuning for parameter-efficient adaptation. CADO achieves state-of-the-art performance across diverse benchmarks, validating that objective alignment is essential for unlocking the full potential of heatmap-based solvers.

We analyse linear ensemble sampling (ES) with standard Gaussian perturbations in stochastic linear bandits. We show that for ensemble size $m=Θ(d\log n)$, ES attains $\tilde O(d^{3/2}\sqrt n)$ high-probability regret, closing the gap to the Thompson sampling benchmark while keeping computation comparable. The proof brings a new perspective on randomized exploration in linear bandits by reducing the analysis to a time-uniform exceedance problem for $m$ independent Brownian motions. This continuous-time lens appears particularly natural here: it yields an exact representation of the relevant discrete-time processes, and we do not know another route to a sharp ES bound.

Online reinforcement learning in non-episodic, finite-horizon MDPs remains underexplored and is challenged by the need to estimate returns to a fixed terminal time. Existing infinite-horizon methods, which often rely on discounted contraction, do not naturally account for this fixed-horizon structure. We introduce a modified Q-function: rather than targeting the full-horizon, we learn a K-step lookahead Q-function that truncates planning to the next K steps. To further improve sample efficiency, we introduce a thresholding mechanism: actions are selected only when their estimated K-step lookahead value exceeds a time-varying threshold. We provide an efficient tabular learning algorithm for this novel objective, proving it achieves fast finite-sample convergence: it achieves minimax optimal constant regret for $K=1$ and $\mathcal{O}(\max((K-1),C_{K-1})\sqrt{SAT\log(T)})$ regret for any $K \geq 2$. We numerically evaluate the performance of our algorithm under the objective of maximizing reward. Our implementation adaptively increases K over time, balancing lookahead depth against estimation variance. Empirical results demonstrate superior cumulative rewards over state-of-the-art tabular RL methods across synthetic MDPs and RL environments: JumpRiverswim, FrozenLake and AnyTrading. Code is provided on \href{https://github.com/jamie01713/K-Step-Lookahead}{github}.

This paper develops a finite-sample statistical theory for in-context learning (ICL), analyzed within a meta-learning framework that accommodates mixtures of diverse task types. We introduce a principled risk decomposition that separates the total ICL risk into two orthogonal components: Bayes Gap and Posterior Variance. The Bayes Gap quantifies how well the trained model approximates the Bayes-optimal in-context predictor. For a uniform-attention Transformer, we derive a non-asymptotic upper bound on this gap, which explicitly clarifies the dependence on the number of pretraining prompts and their context length. The Posterior Variance is a model-independent risk representing the intrinsic task uncertainty. Our key finding is that this term is determined solely by the difficulty of the true underlying task, while the uncertainty arising from the task mixture vanishes exponentially fast with only a few in-context examples. Together, these results provide a unified view of ICL: the Transformer selects the optimal meta-algorithm during pretraining and rapidly converges to the optimal algorithm for the true task at test time.

Integrated Gradients (IG) is a widely used attribution method in explainable AI, particularly in computer vision applications where reliable feature attribution is essential. A key limitation of IG is its sensitivity to the choice of baseline (reference) images. Multi-baseline extensions such as Expected Gradients (EG) assume uniform weighting over baselines, implicitly treating all baseline images as equally informative. In high-dimensional vision models, this assumption often leads to noisy or unstable explanations. This paper proposes Weighted Integrated Gradients (WG), a principled approach that evaluates and weights baselines to enhance attribution reliability. WG introduces an unsupervised criterion for baseline suitability, enabling adaptive selection and weighting of baselines on a per-input basis. The method preserves the core axiomatic properties of IG in a generalized weighted-baseline form. Under an expected, proxy-based fitness--relevance monotonicity assumption, WG provides a probabilistic justification for assigning larger weights to more informative baselines. Experiments on commonly used image datasets and models show that WG improves over EG under our protocol, with up to 36% gains across evaluated convolutional and Transformer architectures. These gains come with additional fitness-evaluation cost, so WG should be viewed as an attribution-fidelity trade-off rather than a faster alternative to EG. By moving beyond the assumption that all baselines contribute equally, Weighted Integrated Gradients offers a clearer and more reliable approach to explaining computer-vision models, improving both understanding and practical usability in explainable AI.

This paper presents Two-Stage LKPLO, a novel multi-stage outlier detection framework that overcomes the coexisting limitations of conventional projection-based methods: their reliance on a fixed statistical metric and their assumption of a single data structure. Our framework uniquely synthesizes three key concepts: (1) a generalized loss-based outlyingness measure (PLO) that replaces the fixed metric with flexible, adaptive loss functions like our proposed SVM-like loss; (2) a global kernel PCA stage to linearize non-linear data structures; and (3) a subsequent local clustering stage to handle multi-modal distributions. Comprehensive 5-fold cross-validation experiments on 10 benchmark datasets, with automated hyperparameter optimization, demonstrate that Two-Stage LKPLO achieves state-of-the-art performance. It significantly outperforms strong baselines on datasets with challenging structures where existing methods fail, most notably on multi-cluster data (Optdigits) and complex, high-dimensional data (Arrhythmia). Furthermore, an ablation study empirically confirms that the synergistic combination of both the kernelization and localization stages is indispensable for its superior performance. This work contributes a powerful new tool for a significant class of outlier detection problems and underscores the importance of hybrid, multi-stage architectures.

We study fine-grained gap-dependent regret bounds for model-free reinforcement learning in episodic tabular Markov Decision Processes. Existing model-free algorithms achieve minimax worst-case regret, but their gap-dependent bounds remain coarse and fail to fully capture the structure of suboptimality gaps. We address this limitation by establishing fine-grained gap-dependent regret bounds for both UCB-based and non-UCB-based algorithms. In the UCB-based setting, we develop a novel analytical framework that explicitly separates the analysis of optimal and suboptimal state-action pairs, yielding the first fine-grained regret upper bound for UCB-Hoeffding (Jin et al., 2018). To highlight the generality of this framework, we introduce ULCB-Hoeffding, a new UCB-based algorithm inspired by AMB (Xu et al.,2021) but with a simplified structure, which enjoys fine-grained regret guarantees and empirically outperforms AMB. In the non-UCB-based setting, we revisit the only known algorithm AMB, and identify two key issues in its algorithm design and analysis: improper truncation in the $Q$-updates and violation of the martingale difference condition in its concentration argument. We propose a refined version of AMB that addresses these issues, establishing the first rigorous fine-grained gap-dependent regret for a non-UCB-based method, with experiments demonstrating improved performance over AMB.

While normalization techniques are widely used in deep learning, their theoretical understanding remains relatively limited. In this work, we establish the benefits of (generalized) weight normalization (WN) applied to the overparameterized matrix sensing problem. We prove that WN with Riemannian optimization achieves linear convergence, yielding an exponential speedup over standard methods that do not use WN. Our analysis further demonstrates that both iteration and sample complexity improve polynomially as the level of overparameterization increases. To the best of our knowledge, this work provides the first characterization of how WN leverages overparameterization for faster convergence in matrix sensing.

We introduce a practical, black-box framework termed Detection Augmented Learning (DAL) for the problem of piecewise stationary bandits without knowledge of the underlying non-stationarity. DAL accepts any stationary bandit algorithm with order-optimal regret as input and augments it with a change detector, enabling applicability to all common bandit variants. Extensive experimentation demonstrates that DAL consistently surpasses all state-of-the-art methods across diverse non-stationary scenarios, including synthetic benchmarks and real-world datasets, underscoring its versatility and scalability. We provide theoretical insights into DAL's strong empirical performance, complemented by thorough empirical validation.

We introduce a class of algebraic varieties naturally associated with ReLU neural networks, arising from the piecewise linear structure of their outputs across activation regions in input space, and the piecewise multilinear structure in parameter space. By analyzing the rank constraints on the network outputs within each activation region, we derive polynomial equations that characterize the functions representable by the network. We further investigate conditions under which these varieties attain their expected dimension, providing insight into the expressive and structural properties of ReLU networks.

Modern machine learning repeatedly manipulates probability measures: empirical datasets, generated samples, latent distributions, class-conditional laws, particle systems, weights of wide networks and attention patterns. Optimal transport is useful in this setting because it compares such objects by asking how mass should move. It therefore combines a statistically meaningful notion of discrepancy with a geometry of interpolation, dual certificates and variational dynamics. This makes OT a common language for losses, generative modeling, domain adaptation, robust learning, barycenters, gradient flows and mean-field descriptions of learning algorithms. This book presents the main OT techniques with these machine-learning uses in mind. It starts from finite assignment and the Monge map viewpoint, passes to Kantorovich couplings and dual potentials, and then explains the algorithmic ideas that make transport usable: linear programming, semi-discrete cells, Sinkhorn scaling and low-dimensional projections. The same objects are then reused as a geometry of measures, giving Wasserstein distances, barycenters, gradient flows, dynamic formulations and Gaussian/Bures formulas. The final chapters emphasize the variants most relevant to modern ML: divergences and adversarial losses, entropic and unbalanced relaxations, robust or spectral ground geometries, Gromov and quantum extensions, and transport-based views of generative models, mean-field networks and attention dynamics. The goal is to keep the mathematics explicit while exposing the computational and geometric intuitions needed to turn OT into a working toolbox for machine learners.

Motivated by real-world applications that necessitate responsible experimentation, we introduce the problem of best arm identification (BAI) with minimal regret. This variant of the multi-armed bandit problem elegantly amalgamates two of its most ubiquitous objectives: regret minimization and BAI. More precisely, the agent's goal is to identify the best arm with a prescribed confidence level $δ$, while minimizing the cumulative regret up to the stopping time. Focusing on single-parameter exponential families of distributions, we leverage information-theoretic techniques to establish an instance-dependent lower bound on the expected cumulative regret. Moreover, we present an impossibility result that underscores the tension between cumulative regret and sample complexity in fixed-confidence BAI. Complementarily, we design and analyze the Double KL-UCB algorithm, which achieves asymptotic optimality as the confidence level tends to zero. Notably, this algorithm employs two distinct confidence bounds to guide arm selection in a randomized manner. Our findings elucidate a fresh perspective on the inherent connections between regret minimization and BAI.

In this paper, we address the classification of instances represented by distributions on a vector space rather than single points. We consider classification algorithms based on pairwise distances, specifically, the Wasserstein metric between distributions. Central to our investigation is dimension reduction within the Wasserstein metric space to enhance classification accuracy. We introduce a novel approach grounded in the principle of maximizing Fisher's ratio, defined as the quotient of between-class variation to within-class variation. The directions in which this ratio is maximized are termed discriminant coordinates or canonical variates axes. In practice, both between-class and within-class variations are defined as the average squared Wasserstein distances between pairs of distributions, with the pairs either belonging to the same class or to different classes. This ratio optimization is achieved through an iterative algorithm, which alternates between optimal transport and maximization steps within the vector space. Empirical studies are conducted to assess the algorithm's convergence; and experimental results demonstrate that the dimension reduction technique substantially enhances classification performance. Moreover, the new method outperforms well-established algorithms that operate on vector representations derived from distributional data. It also exhibits robustness to variations in how instances are summarized by distributions, such as the number of components in a Gaussian mixture model (GMM) representation.

Determining an appropriate sample size for a study is a crucial step in planning scientific research. Appropriate sample size planning avoids both inadequate and inflated sample sizes. Inflated sample sizes wastes resources, time and effort of human subjects, and lives of experimental animals. Inadequate sample sizes, a much more common problem, wastes even more resources through the inability to detect biologically meaningful differences and encourages questionable research practices like $p$-hacking. Microbiome studies are particularly challenged by small sample sizes, particularly in studies of human subjects or expensive animal models. In practice, the statistical power of taxa within a differential abundance study is influenced by the effect size (typically quantified as fold change), mean abundance of individual taxa, and the number of samples. We present a novel approach for sample size calculation for differential abundance studies as a function of effect size, mean abundance and statistical power of taxa. Our method is implemented in the power.nb R package, available at https://michaelagronah.com/power.nb/articles/stub.html. We applied our model for sample size calculation using estimates of mean abundance and fold change of taxa obtained from thirty real-world microbiome datasets. Our results showed that differential abundance microbiome studies require larger sample sizes than are currently prevalent in the literature to achieve adequate statistical power. Our framework will help researchers make informed decisions about appropriate sample sizes.

High-throughput gene expression data exhibit high dimensionality, complex intergene dependence, and pronounced biological heterogeneity across samples, presenting major challenges for unsupervised clustering and disease subtype discovery. We introduce a module-structured mixture factor model that combines finite mixture modelling with low-rank latent factor representations defined at the gene-module level. By explicitly modelling gene modules in both the mean and covariance structure, the proposed framework decomposes expression variability into global gene-specific effects, cluster-specific module-level shifts, latent dependence within modules, and gene-specific residual noise. An Expectation--Conditional Maximisation algorithm is developed for parameter estimation, allowing stable and scalable inference in high-dimensional transcriptomic settings. This framework enables interpretable unsupervised identification of disease-associated molecular subtypes and phenotypic heterogeneity across two autoimmune diseases using a large clinical transcriptomic dataset.

Functional data collected as continuously observed trajectories arise naturally in many biomedical settings, and a key inferential goal is understanding how such functional covariates relate to time-to-event outcomes while allowing that relationship to vary across subgroups defined by scalar characteristics. Existing frequentist approaches to functional accelerated failure-time (AFT) models struggle to flexibly capture the joint, nonlinear influence of scalar covariates and time on the functional effect, and none adequately address the measurement error that frequently contaminates functionally observed exposures. We propose a Bayesian functional AFT model in which the functional coefficient is a varying effect function of both time and a set of scalar covariates, modeled through a Gaussian process single-index structure that provides a flexible, nonlinear framework for subgroup modification. Measurement error in the functional covariate is handled by pairing a proxy observation with an instrumental variable that accommodates non-linear associations with the latent functional exposure. A prominent source of such functional data is wearable devices, which can continuously monitor physical activity (PA) behavioral patterns over time, yet whose outputs are well known to be prone to measurement error and to exhibit heterogeneous associations with health outcomes across demographic subgroups. Through simulations, we show that our approach recovers the true varying functional effects and reduces bias relative tonaïve models that ignore measurement error. We apply our methods to the Reasons for Geographical and Racial Differences in Stroke (REGARDS) study to investigate how step-count physical activity relates to time-to-death from ischemic stroke across racial and regional groups.

Interim monitoring in time-to-event trials must balance inferential maturity with operationally meaningful timing. Event-driven designs align analyses with event accumulation but can produce substantial and unpredictable calendar delays, whereas enrollment-driven designs provide predictable timing but may rely on immature follow-up. We propose the Window-Cohort with Calibrated Follow-Up Requirement (WCR) framework, which directly parameterizes follow-up maturity through a locked cohort size and a post-lock follow-up requirement. The interim analysis is conducted after the prespecified cohort has accrued the calibrated minimum follow-up, while enrollment may continue and later patients are reserved for the final analysis. The framework distinguishes restricted follow-up for landmark survival estimands from unrestricted follow-up for proportional hazards estimands, thereby linking the effective information horizon to the estimand. Design parameters and decision thresholds are jointly calibrated through constrained optimization to control type I error and power while balancing calendar time, interim maturity, and decision-lag burden. Simulation studies motivated by a rare pediatric oncology trial show that WCR attains target operating characteristics under the calibration model and offers more stable and interpretable interim timing than conventional event-driven and enrollment-driven approaches. The methodology is implemented in the open-source R package WCRBayesDesign, available on CRAN. WCR reframes interim monitoring as an information-time alignment problem and provides a practical design strategy for single-arm trials with sparse events, slow accrual, and long-horizon endpoints.

The effective reproduction number ($R_t$) is widely used to track epidemic dynamics in real time. The standard estimation framework uses "instantaneous $R_t$," defined via the renewal equation, which relates new infections to past infections through a generation interval distribution. Compartmental models like SEIR yield a seemingly distinct quantity, "mechanistic $R_t$," based on the effective contact rate and duration of infectiousness. We prove these two definitions are equivalent under homogeneous mixing, the standard assumption in compartmental modeling. We also derive the generation interval distribution implied by SEIR dynamics. A practical consequence is that generation intervals, often treated as assumption-light inputs to renewal equation estimators, in fact encode specific compartmental structure.

In studies with clustered or hierarchical data structures, quantifying between-cluster heterogeneity, referred to as contextual effects, is crucial for valid cluster-level inference. The median odds ratio (MOR), derived from random effects (RE) logistic regression models for clustered binary data, provides an intuitive assessment of contextual effects. Most existing research focuses on point estimation of the MOR for two-level models, with limited exploration of its statistical properties under complex multilevel structures. However, the development of corresponding interval estimators is essential for statistical inference. Moreover, many real-world datasets, particularly those from multistage surveys, involve hierarchical structures beyond two levels, where contextual effects at each level are of interest. This paper discusses the estimation of MOR for both the two-and three-level binary data, with particular emphasis on interval estimation. Since the MOR is a post-estimation measure based on variance components of the RE logit model, its confidence interval is derived using the Delta method, treating the log-transformed MOR as asymptotically normal. The approach is demonstrated across different model specifications in two-and three-level settings. An extensive simulation study evaluated the performance of the MOR estimators across diverse scenarios in hierarchical data settings. The results showed that the estimators exhibited negligible bias and satisfactory coverage probability of a 95% confidence interval for moderate to large samples, with small-sample bias mainly due to variance component estimation. An application of the methods for estimating the contextual effect on C-section delivery demonstrated that the proposed framework enhances interpretability and supports more informed statistical and policy-oriented analyses.

Traditional epidemic models frequently assume behavioral homogeneity. The susceptible-infected-recovered model provides a robust foundation for characterizing disease transmission, but it does so without accounting for how people actually respond to risk. In contrast, behavioral change models incorporate mechanisms that capture how individuals adjust their actions during an outbreak, recognizing that rising infection risk typically motivates protective behaviors. Yet both approaches share a key limitation: they overlook the inherent heterogeneity of a population. In reality, communities are a complex mixture of risk tolerances and behavioral tendencies. Ignoring this inherent heterogeneity can obscure important differences in how individuals perceive and respond to disease threats. This paper introduces a novel Bayesian mixture model designed to address this limitation by partitioning the population into two distinct behavioral patterns: risk-neutral individuals, who maintain baseline contact rates, and risk-averse individuals, who modulate their behavior in response to epidemic severity. By integrating these disparate dynamics into a unified transmission framework, the proposed model explicitly accounts for varying population behaviors often overlooked by aggregate approaches. Through simulation studies and empirical data applications, we demonstrate that this approach significantly outperforms traditional models in parameter recovery, epidemic trajectory estimation, and forecasting precision. The findings suggest that failing to account for behavioral diversity leads to biased peak estimates and artificially stretched epidemic curves. Consequently, this research provides a more nuanced computational toolkit for predicting outbreak trajectories in socially fragmented environments, ensuring that public health intervention strategies are informed by a foundation of behavioral realism.

Penalized generalized estimating equations (PGEE) stabilize point estimation for longitudinal binary data under near-separation, but inference still depends on how the sandwich variance is corrected. Existing corrections for PGEE can overadjust in high-leverage directions, require restrictive pooling assumptions, or add global regularization without explaining the bias. We establish first-order asymptotics for PGEE along convergent interior-root sequences and derive a matrix characterization of the parameter-specific overcorrection induced by full leverage adjustment. Finite-sample calibration is limited by both mean bias and the variability of leverage-corrected variance estimates. We propose $\hat{V}_{AR}$, which keeps the score-level leverage correction and adds a finite-sample upward translation dominated at first order by the finite-population factor, with a smaller centering term. In simulations, $\hat{V}_{AR}$ gives conservative or near-nominal type I error in low-event, small-$N$ settings, including $N = 10$, where several standard corrections remain anti-conservative and pooling estimators are unavailable for unbalanced designs.

We propose an online false discovery rate (FDR) controlling method based on conditional local FDR (LIS), designed for infectious disease datasets that are discrete and exhibit complex dependencies. Unlike existing online FDR methods, which often assume independence or suffer from low statistical power in dependent settings, our approach effectively controls FDR while maintaining high detection power in realistic epidemic scenarios. For disease modeling, we establish a Dynamic Bayesian Network (DBN) structure within the Susceptible-Infected-Susceptible (SIS) model, a widely used epidemiological framework for infectious diseases. Our method requires no additional tuning parameters apart from the width of the sliding window, making it practical for real-time disease monitoring. From a statistical perspective, we prove that our method ensures valid FDR control under stationary and ergodic dependencies, extending online hypothesis testing to a broader range of dependent and discrete datasets. Additionally, our method achieves higher statistical power than existing approaches by leveraging LIS, which has been shown to be more powerful than traditional $p$-value-based methods. We validate our method through extensive simulations and real-world applications, including the analysis of infectious disease incidence data. Our results demonstrate that the proposed approach outperforms existing methods by achieving higher detection power while maintaining rigorous FDR control.

We introduce biarchetype analysis for the first time in the context of univariate functional data. This unsupervised methodology extends archetype analysis by simultaneously identifying archetypal structures across both the cases (countries, in our application) and the temporal argument. Both cases and time points are expressed as mixtures of biarchetypes, yielding a concise and highly interpretable representation of complex functional observations. Although biarchetype analysis is not intended as a clustering technique, it offers superior interpretability compared with biclustering approaches, as it is based on extreme, representative patterns rather than average centroids, thereby enhancing human comprehension. We apply the proposed method to 10-year government bond yields of European countries over the period 2001-2025. The results identify three distinct time regimes (the pre-crisis period, the euro-area sovereign debt crisis, and the post-crisis period), and reveal Germany, Greece, and Hungary as country archetypes.

A recent methodology for extracting archetypal profiles from three-way asymmetric proximity data is applied to a dataset comprising 23 x 23 x 3 two-mode, three-way asymmetric dissimilarity matrices. The asymmetric dissimilarities are based on the oriented wavelet squared coherence of 10-year government bond yields among 23 European countries across three time intervals from 2001 to the present. First, the h-plot is computed for the unconditional two-mode three-way data, which is then represented in a unified Euclidean space, providing an intuitive and interpretable visualization. Subsequently, archetypoid analysis is performed. This unsupervised methodology identifies the archetypal countries and expresses all remaining countries as mixtures of these archetypal instances. Additionally, the degree of asymmetry for each country is calculated, offering further insight into the structure of the dissimilarities. The dataset and code are provided to support reproducible research.

We introduce the Multiplex Network Hawkes model, which extends the network Hawkes framework of Linderman & Adams (2014) by allowing multiple excitation layers whose weights depend on observed edge and node covariates. We use the model to investigate how contagion in financial networks is affected by different transmission channels. The multiplex structure separates channel-specific contributions within a single inferred transmission network, allowing candidate propagation mechanisms to be compared directly rather than being absorbed into one homogeneous excitation layer. Covariate-dependent excitation allows us to investigate sources of transmission. We make posterior inference about the inferred directed network and its excitation dynamics using an MCMC sampler. The application uses a broad cross-industry credit default swap (CDS) dataset of 99 North American and European firms, including banks, insurers and non-financial firms over 2004-2022. We evaluate three candidate contagion channels associated with asset similarity, solvency and profitability. The results indicate sparse contagion pathways, with systemic-risk transmission concentrated in outward flows from a small number of influential institutions rather than in mutual feedback between institutions. The channel results show that industry similarity is the most consistently supported asset-similarity effect, while aggregate layer contributions indicate that asset-similarity, solvency and profitability channels all contribute to inferred excitation.

This study examines whether machine learning (ML) models can outperform the naive random walk benchmark in forecasting the monthly USD/CAD exchange rate. Using daily data from the Bank of Canada spanning January 2017 to May 2026, resampled into 113 monthly observations, five ML models are evaluated: linear regression, random forest, gradient boosting, XGBoost, and AdaBoost. These models are benchmarked against the naive random walk model and exponential smoothing with Holt-Winters seasonality (ETS). All models are evaluated using an expanding-window framework to maintain strict out-of-sample integrity, and forecast-accuracy differences are assessed using the Diebold-Mariano (DM) test. Structural break detection identifies four significant breakpoints in the series, corresponding to the escalation of the US-China trade war in 2018, the COVID-19 economic recovery in 2020, the peak of the Bank of Canada rate-hiking cycle in 2022, and the start of the Bank of Canada rate-cutting cycle in 2024. SHAP, or Shapley Additive Explanations, analysis is applied to interpret the drivers of the best-performing ML model. The results show that the naive random walk model remains a formidable benchmark. Linear regression is the only model that statistically outperforms the naive random walk model, with a DM statistic of 3.0585 and a p value of 0.0071, whereas the ML ensemble models show only marginal differences. Random Forest with an expanding-window framework achieves the lowest MAPE of 1.17 percent among all models except the random walk. SHAP analysis confirms that short-term lags, particularly lag1 and lag2, and recent rolling means dominate predictions, consistent with the near-random-walk behavior of exchange rates.

In the archetype discovery problem a researcher wants to summarize N heterogeneous policy effects of interest that vary over a discrete set of covariates. The goal is to partition the set of covariates into K<N groups -- the archetype sets -- and to provide a summary of the policy effects for each group. We use decision theory to show that, under a weighted mean-squared-error criterion, a procedure analogous to the Sorted Group Average Treatment Effects (GATES) solves the archetype discovery problem. The key difference is that, in the optimal procedure, archetype sets are obtained by weighted K-means clustering of the N heterogeneous policy effects, instead of relying on K equally-spaced quantiles. We show that the procedure that minimizes average risk for a given prior can be obtained by clustering the different values of the posterior mean estimate of the policy effects of interest. Similarly, an approximately minimax procedure in large samples can be obtained by clustering a consistent estimator of the policy effects. In both of these cases, an exact solution to the weighted K-means clustering problem can be found using a simple and well-known dynamic programming algorithm.

Two communities that rarely cite each other -- spatial statisticians fitting high-dimensional weather fields, and quantitative investors building portfolios -- have independently arrived at the same mathematical object: a Schur complement, damped by one interpretable parameter. In spatial modeling the Schur complement is the conditional covariance that makes a Gaussian (Vecchia) pseudo-likelihood estimable at scale, and recent work regularizes it by shrinking toward a base model. In allocation it is the residual risk of a bet net of its hedge, and the same parameter interpolates hierarchical risk parity and the minimum-variance portfolio. We show these are one operation -- reliability shrinkage of a conditional Gaussian -- so that the damping a weather model needs to remain estimable when stations outnumber observations is, term for term, the damping a portfolio needs to remain stable when assets outnumber returns. The optimal amount is a closed-form reliability, a James-Stein shrinkage that is simultaneously a Ledoit-Wolf intensity. The shrinkage machinery is classical, but the identity appears to be new: to our knowledge neither literature has noted that the conditional shrinkage a spatial model fits and the diversification-variance tilt a portfolio chooses are one and the same quantity. We make the correspondence precise, note that the two literatures have each supplied what the other lacks, and report a small experiment on the one genuinely open choice -- how to set the damping -- suggesting the spatial community's fitted intensity is, if anything, the better recipe.

Governments with limited fiscal and administrative capacity need to know which Sustainable Development Goals (SDGs) propagate progress through the goal system and how quickly. We map the directed interdependence structure of all seventeen goals using a balanced panel of 114 countries observed annually from 2000 to 2024. The goal series are persistent, trending, and cross-sectionally dependent, so we apply two estimators matched to this regime: a Dumitrescu-Hurlin panel Granger non-causality test, run on first-differenced series, to recover the directed interaction network, and panel local projections with Driscoll-Kraay standard errors to measure the dynamic magnitude of 31 theory-derived indicator linkages. Of 272 directed goal pairs, 84 linkages survive false-discovery control (40 synergies, 44 trade-offs; network density 0.31). Synergies and trade-offs occur at comparable strength, so no single goal behaves as a universal accelerator, and the goal-level hierarchy itself is fragile. Driver-receiver rankings correlate weakly across lag orders and centrality metrics, and under a country bootstrap only two roles are distinguishable from zero: peace and strong institutions as the clearest net receiver, and poverty reduction as the most probable effect-size-weighted driver. The supported linkages are dynamic, accruing over four to five years: sanitation and poverty improvements are the strongest predictors of lower child mortality, and the education-child-health association is corroborated in independent World Development Indicators data across 183 countries. These results caution against rankings-based accelerator policy and support adaptive portfolios built on supported, time-lagged linkages monitored through constituent indicators.

U.S. Treasury yields are central to global asset pricing but are noisy and subject to policy uncertainty, supply-demand forces, and behavioral effects, exposing forecast users to downside risk. We formulate yield curve forecasting as a decision problem under distributional uncertainty and propose a distributionally robust ensemble framework that combines parametric factor models with machine-learning forecasts. A factor-augmented Dynamic Nelson-Siegel model captures yield-curve dynamics, while Random Forests model nonlinear interactions. Robust forecast combinations penalize tail risk and improve out-of-sample performance across maturities. The framework supports disciplined $DV01$-based interest-rate risk management for corporate, institutional and balance-sheet decision makers.

This paper examines the relationship between the genderedness of authors' first names and citation distributions in scholarly production. Merging a first name genderedness table derived from Wikidata with bibliometric data from articles by US-affiliated authors published between 2010 and 2019 and indexed in the Web of Science, we develop a relative distributional framework that compares name, article, and citation counts along a continuous genderedness spectrum. Results show that the lexical structure of the corpus, as captured by the relationship between unique first names (types) and the number of their occurrences (tokens), proves highly stable across author roles. Productivity analyses reveal that disciplinary groups diverge substantially in the direction of the imbalance between femininely- and masculinely-gendered names, with physical sciences showing a consistent masculine skew and social sciences a tendency toward the feminine end of the spectrum. Strikingly, however, the amplitude of distributional divergence remains relatively stable across disciplinary groups, in contrast to its substantial variations in direction. Citation analyses reveal a pervasive citation deficit for femininely- and neutrally-gendered names across all disciplinary groups and author roles. This asymmetry is most consistent across author roles in the life sciences, and absent in the physical sciences except among middle authors, whose unparalleled share plausibly reflects the citation dynamics of large collaborative structures. Overall, although the data and research design support associative rather than causal claims, the trends they reveal are nonetheless consistent with the hypothesis that first name genderedness influences citation recognition through implicit bias operating in low-deliberation evaluative contexts.

U.S. state education agencies mark schools displaying achievement gaps between demographic subgroups as needing improvement. Some schools may have few students in these subgroups, such that average end-of-year test scores only noisily measure the average "true" score-the score one would expect if students took the test many times. This, in addition to the masking of small subgroup averages in publicly available assessment data, poses challenges for evaluating interventions aimed at closing achievement gaps. We introduce propensity score estimates designed to achieve balance on subgroup average true scores. These estimates are available even when noisy measurements are not and improve overlap compared to those that ignore measurement error, leading to greater bias reduction of matching estimators. We demonstrate our methods through simulation and an application to a statewide initiative in Texas for curbing summer learning loss.

This paper introduces a novel ranking of statistical experiments, the linear-Blackwell (LB) order, which can equivalently be characterized by (i) the dispersion of the induced posterior and likelihood ratios in the sense of the linear convex order, (ii) the size of the Lorenz zonoid (the set of statewise expectation profiles), or (iii) the variability of the posterior mean. We apply the LB order to compare experiments in binary-action decision problems and in decision problems with quasi-concave payoffs, as analyzed by Kolotilin, Corrao, and Wolitzky (2025). We also use it to compare experiments in moral hazard problems, building on Holmström (1979) and Kim (1995), and in screening problems with ex post signals.

The rapid adoption of generative AI and Large Language Models (LLMs) has spurred interest in synthetic data as a privacy-preserving alternative to sensitive real-world datasets. However, generating high-utility synthetic data often carries the risk of memorizing and regurgitating private information from the training corpus. In this work, we present a customizable empirical auditing framework designed to detect and explain such data disclosures. Our framework introduces a mechanism to distinguish between "true disclosures"-where the system directly reproduces a user's information-and "phantom disclosures''-where the system incidentally generates a user's data. By partitioning input data into training and holdout sets and applying rigorous statistical hypothesis testing, we determine if observed disclosures are consistent with strict privacy baselines, such as zero-learning or specific Differential Privacy (DP) bounds. Crucially, this approach requires no model access, no canary insertion, and no reference model training -only the synthetic output and a held-out control set. We demonstrate that this framework effectively functions as a membership inference attack, providing empirical lower bounds on privacy leakage that are tighter than prior data-based auditing methods. Our approach is model-agnostic, applies to any synthetic data generation mechanism, and requires orders of magnitude fewer computational resources than shadow-model or canary-based alternatives.

Fairness metrics are essential for rigorously defining, quantifying, and mitigating biases in predictive models. While most existing metrics focus on binary classification tasks, fairness in time-to-event analyses has received limited attention. To address this gap, we propose a novel group fairness metric, the group-conditional Concordance Index (xCI), which extends Harrell's Concordance Index (CI) by conditioning on group membership. The xCI measures both within-group and cross-group ranking accuracy in the presence of right-censored data. We formally define the xCI, prove that CI is a weighted average of xCIs across all possible group pairs, and develop a consistent estimator using inverse probability of censoring weights (IPCW). We further investigate the relationship between xCI and predicted risk scores through analytical derivations and simulation studies. To demonstrate its practical utility, we present two case studies: (i) assessing the fairness of survival models trained on harmonized data from the Framingham Offspring, MESA, and ARIC studies, and (ii) evaluating fairness in existing cardiovascular disease (CVD) risk prediction models using Truveta, a large-scale electronic health record (EHR) database. Our results show that xCI effectively detects biases across demographic groups that are overlooked by existing metrics. Overall, xCI provides a valuable tool for fairness assessment in survival analysis, particularly in constrained resource allocation settings, and complements existing fairness evaluation approaches.

The increasing demand for access to microdata in official statistics and data-intensive applications raises important challenges concerning disclosure risk, inferential validity and preservation of statistical utility. This paper proposes an interpretable energy-driven framework for privacy-aware synthetic data generation in mixed-type data. The proposed methodology combines discriminative modelling, Bayesian-Network proposal mechanisms, Metropolis--Hastings sampling and post-generation optimization within a constrained probabilistic framework. Unlike perturbation-based approaches, privacy-aware behaviour is achieved through constrained stochastic exploration guided by explicit plausibility, privacy, diversity and structural-coherence penalties. The framework is specifically designed for mixed-type tabular data characterized by sparse configurations, heterogeneous variable types and complex multivariate dependency structures. The generation process is formulated as a multi-objective sampling problem balancing statistical fidelity and disclosure-risk while preserving predictive utility. An extensive empirical evaluation is conducted using a mixed-type individual-level dataset containing demographic, behavioural and health-related variables. The validation strategy combines statistical fidelity diagnostics, predictive analyses, diversity measures, nearest-neighbour risk analysis, membership inference attacks and Split Conformal Prediction. The empirical results suggest that the proposed framework is capable of preserving a substantial portion of the predictive and multivariate structure of the original data while limiting exact memorization phenomena and maintaining favourable privacy-aware behaviour. The proposed methodology provides an interpretable framework for synthetic data generation under competing utility and privacy constraints.

Foundation models are now used in settings where the prompts they receive can change quickly. Users change, topics change, policies change, and the model may suddenly face a kind of request that was rare in the calibration data. This makes fixed calibration risky. Conformal prediction and conformal risk control give model-agnostic ways to control error, but they work best when the calibration data still look like the future data. This paper develops PromptShift CRC, a drift-aware conformal risk control method for foundation-model outputs under prompt and domain shift. The method embeds prompts and responses, measures how far the current prompt stream has moved from the calibration pool, gives more weight to relevant or recent calibration examples, and updates the risk level online after observed violations. It reports three practical diagnostics: realized risk error, prompt drift, and effective calibration size. We give conditions under which the method controls risk up to terms for distribution mismatch and weighted quantile uncertainty. In a synthetic prompt-shift benchmark, static conformal risk control fails sharply after drift, while PromptShift-CRC gives the best coverage among the adaptive baselines considered. We then evaluate the same calibration layer on public benchmark derived streams for question answering, toxicity, summarization factuality, and long-context hallucination risk

Continuous evaluation of LLM products relies on a strong LLM judge treated as ground truth: a cheap monitor scores every interaction and a team is paged when the score drifts down. But the judge is itself a model behind an API, and a silent version bump or scoring-prompt update changes how it scores -- so every drift alarm is ambiguous between a worse product and a changed judge. We resolve the ambiguity with a fixed, human-labeled anchor set that the current judge re-scores at a steady interleave, a second betting e-process on the judge-versus-human gap, and a guard-window rule returning a verdict in {none, system, judge}. We prove anytime-validity, one-way identification (only the judge can move the anchors), an attribution race whose design law is that the anchors must out-run the main process they guard, and process orthogonality. On two real judge changes, a silent version bump is detected as judge drift in 60/60 runs with zero judge-to-system misattribution, and a contaminating strict-prompt change is correctly attributed on 110 of 120 runs at guard width 300 -- while the industry-default rolling z-test false-alarms on 75% of drift-free streams. Every experiment replicates on a second domain (TL;DR summarization) with nothing re-tuned, and where the domains differ the differences are the ones the race predicts: the strict-prompt change shifts scores harder there, so the anchors fire faster and attribution becomes perfect (240/240). The monitor runs at approximately 0.64 of the cost of strong-judging every item, or 0.21 in a cheaper-but-deafer regime.

We consider the problem of uncertainty quantification for a pretrained classification model deployed under unknown distribution shift. We propose Audited Conformal Prediction (ACP), a method that leverages a small labeled dataset from the target population to train an auxiliary audit model identifying inputs where the legacy model is likely to fail. By integrating the audit model's outputs into the conformal prediction framework, ACP produces prediction sets that guarantee marginal coverage while achieving substantially higher conditional coverage in practice than existing approaches. We develop and analyze two complementary integration strategies -- one targeting marginal coverage with improved conditional performance, the other providing explicit group-conditional coverage guarantees -- and establish theoretical guarantees for both. Experiments on synthetic and real-world datasets validate the method and illustrate trade-offs between prediction set size and conditional coverage.

Fairness has become an important concern in insurance pricing as insurers increasingly rely on machine learning models to predict expected losses. At the same time, regulatory and privacy constraints often restrict insurers' ability to access or use sensitive attributes such as gender or race. Recent actuarial research addresses fairness in this context through the concept of the discrimination-free premium, which removes both the direct and indirect effects of sensitive attributes while preserving actuarial consistency. However, implementing this approach typically requires access to the sensitive attributes themselves, which may not be available in practice. This paper studies the estimation of discrimination-free insurance premiums when sensitive attributes are observed only in privatized or noise-perturbed form. We consider a multi-party data setting in which insurers observe non-sensitive attributes and outcomes, while a trusted third party holds privatized sensitive attributes generated through a privacy mechanism. Within this framework, we develop statistical methods for estimating discrimination-free premiums using only the privatized attributes. We study two settings of practical relevance: when the privacy mechanism is known and when its noise level is unknown. For both cases, we establish theoretical guarantees for the proposed estimators. Numerical experiments and empirical applications demonstrate that the proposed approach enables fair insurance pricing while respecting privacy and regulatory constraints.

In today's data-driven world, the proliferation of publicly available information raises security concerns due to the information leakage (IL) problem. IL involves unintentionally exposing sensitive information to unauthorized parties via observable system information. Conventional statistical approaches rely on estimating mutual information (MI) between observable and secret information for detecting ILs, face challenges of the curse of dimensionality, convergence, computational complexity, and MI misestimation. Though effective, emerging supervised machine learning based approaches to detect ILs are limited to binary system sensitive information and lack a comprehensive framework. To address these limitations, we establish a theoretical framework using statistical learning theory and information theory to quantify and detect IL accurately. Using automated machine learning, we demonstrate that MI can be accurately estimated by approximating the typically unknown Bayes predictor's log-loss and accuracy. Based on this, we show how MI can effectively be estimated to detect ILs. Our method performs superior to state-of-the-art baselines in an empirical study considering synthetic and real-world OpenSSL TLS server datasets.

AI safety is a rapidly growing area of research that seeks to prevent the harm and misuse of frontier AI technology, particularly with respect to generative AI (GenAI) tools that are capable of creating realistic and high-quality content through text prompts. Examples of such tools include large language models (LLMs) and text-to-image (T2I) diffusion models. As the performance of various leading GenAI models approaches saturation due to similar training data sources and neural network architecture designs, the development of reliable safety guardrails has become a key differentiator for responsibility and sustainability. This paper presents a formalization of the concept of computational safety, which is a mathematical framework that enables the quantitative assessment, formulation, and study of safety challenges in GenAI through the lens of signal processing theory and methods. In particular, we explore two exemplary categories of computational safety challenges in GenAI that can be formulated as hypothesis testing problems. For the safety of model input, we show how sensitivity analysis and loss landscape analysis can be used to detect malicious prompts with jailbreak attempts. For the safety of model output, we elucidate how statistical signal processing can be used to detect AI-generated content. Finally, we discuss key open research challenges, opportunities, and the essential role of signal processing in computational AI safety.

The R package hyreg2 introduces a frequentist framework for estimating latent class models for mixed outcome types using a joint likelihood approach. The method combines continuous and dichotomous data under the assumption that both outcome types arise from a common underlying data-generating process. In the implemented model, continuous responses are assumed to follow a normal distribution, while dichotomous responses are modeled using a binomial distribution. Such models are used in various scientific disciplines to estimate a common set of parameters across different types of data (e.g. clinical trials, econometrics and health economics). Latent class estimation is performed using the expectation-maximization algorithm as implemented in the widely used R package flexmix. The hyreg2 package offers a user-friendly implementation of this joint likelihood framework, allowing users to estimate models without explicitly programming the likelihood function. Heteroskedasticity as well as censored data can be taken into account. In addition to model estimation, the package provides dedicated summary and visualization functions to facilitate the interpretation of results. The article presents the methodological framework underlying the package and illustrates its functionality through an example based on the estimation of an EQ-5D-5L value set.

Generalized linear mixed models (GLMMs) are widely used for analyzing correlated data, such as longitudinal and multilevel data. With over 15 $\texttt{R}$ packages available on $\texttt{CRAN}$ for fitting GLMMs, practitioners face a difficult choice regarding which package yields accurate estimates, converges reliably, and offers reasonable computational speed. Existing comparisons are either limited to methods within a single package or focus on narrow criteria such as speed alone. To address this gap, we systematically compared seven representative $\texttt{R}$ packages -- $\texttt{lme4}$, $\texttt{GLMMadaptive}$, $\texttt{glmmTMB}$, $\texttt{MASS}$, $\texttt{hglm}$, $\texttt{brms}$, and $\texttt{rstanarm}$ -- that implement different estimation frameworks. By using Monte Carlo simulations across 24 scenarios, we evaluated each package in terms of convergence ratios, computational time, estimation accuracy, and hypothesis testing performance. Our results showed that $\texttt{lme4_AGQ}$ and $\texttt{GLMMadaptive}$ yield the highest accuracy and convergence ratios, although $\texttt{GLMMadaptive}$ becomes slower under complex random-effect structures. $\texttt{lme4_LA}$ and $\texttt{glmmTMB}$ are computationally fast but exhibit lower convergence ratios and larger bias, especially for variance components. $\texttt{MASS}$ and $\texttt{hglm}$ are also fast, but $\texttt{MASS}$ yields liberal univariate tests and $\texttt{hglm}$ lacks support for correlated random effects and multivariate testing. Between two Bayesian packages, $\texttt{rstanarm}$ converges reliably and produces valid univariate tests, whereas $\texttt{brms}$ is extremely slow, limiting its practical utility. Based on these findings, we provide practical recommendations for choosing GLMM tool in applied research.

A significant gap exists between theory and practice in deep learning. Generalization and approximation error bounds are often derived for simplified models or are too loose to be informative. Many rely on the manifold hypothesis and on geometric regularity such as intrinsic dimension, curvature, and reach. Progress requires insight into data-manifold geometry and suitable benchmarks, yet existing options are polarized: analytic manifolds with known geometry but limited applicability, or real-world datasets where geometry is only coarsely estimable. We introduce a benchmarking framework for studying data geometry. We repurpose and extend dSprites and COIL-20 with additional transformation dimensions and dense, axis-aligned sampling, and pair them with finite-difference estimators that recover curvature, reach, and volume at near-ground-truth accuracy in a regime where general-purpose estimators are unreliable or difficult to deploy. The framework is intended as a controlled testbed, useful as a calibration environment for geometric estimators and a sandbox for probing theoretical assumptions. To illustrate its use, we present two application studies, namely assessing the scaling behavior of the bounds of Genovese et al. and Fefferman et al., and tracking the layer-wise geometry of a $β$-VAE, highlighting the behavior of current bounds and the value of controlled benchmarks for guiding and validating future theory. A reference implementation is available at https://github.com/koulakis/manifold-microscope.

Industrial retrofit planning depends on structured operational data rather than free text: planners must estimate whether a newly registered prototype will require a retrofit, which retrofit package it will need, and how long the work will take. We study an industrial dataset linking a prototype-registration system (284,271 vehicles) with a retrofit-management system (48,716 cleaned visits), and compare strong tabular machine learning baselines with three LLM-based strategies on row-serialized inputs: embedding features (Amazon Titan), direct prompted classification (Claude Sonnet 4), and an ML+LLM stacking approach. Across binary occurrence prediction, 15-way retrofit-type classification, per-visit duration regression, and an aggregated monthly benchmark, classical tree ensembles remain the strongest standalone models. However, the LLM results reveal a consistent pattern: embeddings remain useful on tables (binary AUC = 0.982), direct prompting collapses once semantic signal is stripped by hashing (binary AUC = 0.500; multiclass weighted F1 = 0.018), and hybrid stacking yields the best manually built multiclass model (weighted F1 = 0.626). On the monthly benchmark, lag-based machine learning outperforms time-series foundation models, though Chronos-small remains competitive in zero-shot forecasting. The results suggest that on privacy-constrained industrial tables, LLMs are more effective as complementary components than as replacements for strong tabular baselines.

We propose a Koopman-enhanced physics-informed neural network (K--PINN) framework for parameter inference and forecasting in nonlinear epidemic models. This method combines Koopman operator theory and physics-informed learning. It maps epidemic states into a latent observable space where the dynamics evolve approximately linearly while satisfying the governing epidemic equations through automatic differentiation. This integration improves interpretability, parameter identifiability, and long-term predictive stability. We apply the proposed framework to a normalized SEIRSD epidemic model and evaluate it using synthetic monkeypox (Mpox) data and real-world datasets from Germany, Morocco, and Sweden for the SARS-CoV-2 virus. Synthetic trajectories are generated using a structure-preserving, nonstandard finite difference scheme to ensure reliable training data. Numerical results demonstrate that K--PINN achieves more accurate parameter estimation, trajectory reconstruction, and long-term forecasting than classical PINNs and Koopman-EDMD approaches. These results suggest that K--PINN is an effective machine learning framework for epidemic modeling that can be extended to more complex systems.

Accurate passenger queue forecasting in airport terminals is essential for efficient departure operations, as it enables proactive congestion management. However, time-varying passenger demand and heterogeneous facility usage across multiple departure facilities make forecasting challenging. In this work, we propose a passenger queue forecasting framework that learns historical passenger flow patterns from operational data. The proposed model employs a Transformer-based architecture to capture temporal dependencies and inter-facility correlations using past queue length and waiting time at departure gates and security checkpoints, together with passenger throughput at check-in islands. The learned representations are mapped to two facility-specific prediction heads to predict queue length and waiting time at departure gates and security checkpoints. Experimental results demonstrate accurate forecasts up to two hours ahead. The proposed approach offers practical real-time decision support for proactive queue management and staff reallocation in airport terminal operations.

Crop rotation impacts on soil nutrients are typically assessed using field-averaged or single-nutrient analyses that ignore spatial heterogeneity and multivariate interactions. We propose a multivariate lattice model treating soil as a 4D tensor (space, time, and N, P, K channels). Crop rotations are represented as force vectors, with soil buffering capacity ("stiffness") varying spatially with texture. Lateral nutrient movement is introduced via kernel smoothing. Cumulative impact is quantified by Euclidean distance in N-P-K space, with significance assessed via Cramer-von Mises permutation tests. Simulating a three-year corn-soybean-wheat rotation on a 20 x 20 heterogeneous grid shows mean stress of 0.63 after one cycle, with maximum 0.91 in sandy areas. Phosphorus depletion (17.9%) exceeds nitrogen (10.8%), dominating stress in 19% of cells - obscured by single-nutrient analyses. Continuous corn increases mean stress by 41%. Cramer-von Mises tests detect significant deviation (p <= 0.002), and Moran's I (0.29-0.30) confirms spatial autocorrelation. Our framework identifies risk zones and guides site-specific management, bridging geostatistics with mechanistic crop models.

In this paper we develop a deforestation detection pipeline that incorporates optical and Synthetic Aperture Radar (SAR) data. A crucial component of the pipeline is the construction of anomaly maps of the optical data, which is done using the residual space of a discrete Karhunen-Loéve (KL) expansion. Anomalies are quantified using a concentration bound on the distribution of the residual components for the nominal state of the forest. This bound does not require prior knowledge on the distribution of the data. This is in contrast to statistical parametric methods that assume knowledge of the data distribution, an impractical assumption that is especially infeasible for high dimensional data such as ours. Once the optical anomaly maps are computed they are combined with SAR data, and the state of the forest is classified by using a Hidden Markov Model (HMM). We test our approach with Sentinel-1 (SAR) and Sentinel-2 (Optical) data on a $92\,km \times 92\,km$ region in the Amazon forest. The results show that both the hybrid optical-radar and optical only methods achieve high accuracy that is superior to the recent state-of-the-art hybrid method. Moreover, the hybrid method is significantly more robust in the case of sparse optical data that are common in highly cloudy regions.

Official statistics are undergoing a significant transformation, as national statistical institutes transition from traditional single-source data production systems to integrated systems of statistical registers combining administrative, census, and survey data. The resulting multisource register-based estimates are prone to multiple interacting sources of error, yet rigorous scalable frameworks for quantifying their accuracy remain underdeveloped. This work discusses and validates a global measure of error assessment for such multisource register-based statistics. Focusing on two central sources of uncertainty, sampling and modelling, we derive an analytical solution that accurately approximates the global error of mass-imputation procedures under a multinomial logistic model. The proposed measure is interpretable, flexible, and computationally scalable, enabling on-the-fly accuracy quantification for user-defined, unplanned domain-specific statistics on population totals. Its validity is established theoretically and confirmed through simulation studies. An application to education data from the Italian National Institute of Statistics is presented.

A central objective of machine learning is to identify structure and patterns in data. Advances in data acquisition have increasingly produced datasets whose observations possess rich geometric form, giving rise to shape spaces that encode variability in object geometry. Such datasets arise across a wide range of disciplines, including biology, medicine, anthropology, and computer vision, where subtle geometric differences often carry important scientific information. Traditional machine learning methods, however, are frequently ill-equipped to account for the nonlinear geometric structure underlying these data. This survey synthesizes a rapidly growing body of work on shape space analysis, which provides a mathematical and computational framework for the study of geometric data. Drawing on ideas from differential geometry, statistics, and machine learning, we organize the literature around a common analytical pipeline: shape representation and parameterization, the rigorous construction of robust geodesic metrics, statistical analysis on shape spaces, and geometry-aware learning methods. We discuss how these tools enable the characterization of shape variability, the comparison of geometric objects, and the analysis of structural trajectories across populations and time. To illustrate the breadth of the field, we highlight applications spanning multiple scales of biological organization, including studies of subcellular morphology and primate tooth evolution. Across these and many other domains, researchers face common challenges arising from complex, nonlinear, and often unaligned geometric variation. The review concludes by identifying key theoretical and computational challenges, as well as emerging opportunities driven by increasingly large and diverse geometric datasets.

We study the problem of detecting a faint geometric signal hidden in an otherwise random graph. Formally, we consider a hypothesis testing problem in which, under the null, the observed graph is an Erdős--Rényi random graph $\mathcal{G}(n,q)$, while under the alternative a random geometric graph $\mathcal{G}(k,q,d)$ is planted on $k\le n$ vertices. The planted subgraph is generated from independent random points on the unit sphere $\mathbb{S}^{d-1}$, with edges determined by latent geometric proximity and calibrated to have edge density $q$. Our goal is to characterize the statistical and computational limits of detecting this hidden geometry. We derive sharp information-theoretic lower bounds that identify regimes where detection is impossible and provide algorithms that achieve these limits whenever detection is feasible. We further investigate the computational complexity of the problem and determine when efficient polynomial-time tests exist. The model exhibits an \emph{easy--hard--impossible} phase transition: some regimes allow efficient detection, others permit detection only with computationally intractable procedures, and still others render detection impossible even with unlimited computational power. As evidence for the computational barrier, we prove that all low-degree polynomial algorithms fail throughout the conjecturally hard regime, demonstrating a sharp gap between statistical and computational feasibility.

We study Euler characteristics of scaled toric varieties arising from second hypersimplices. In algebraic statistics, these are closely connected to maximum likelihood (ML) degrees of toric models. We establish a correspondence between delta-matroids and the non-vanishing factors of the principal $A$-determinant, providing an explicit connection between delta-matroid theory and algebraic statistics. Using this framework, we show that a conjectured minimum ML degree is realizable by a suitable embedding of the variety. Furthermore, for second hypersimplices up to order six, we prove that this value is minimal among all embeddings, as conjectured by Clarke et al. (2024).

The Brody distribution, originally a phenomenological interpolation between Poisson and Wigner level-spacing statistics in quantum chaos, is calibrated here as a quantitative measure of short-range exclusion in 2D spatial point processes. Two results form the core. First, the 2D complete-spatial-randomness baseline is recalibrated to $β=0.96\pm0.15$, correcting the inappropriate 1D Poisson reference. Second, an empirical $β$--$r_{\text{excl}}$ calibration is validated against the effective hard-core radius with Spearman $ρ=0.988$. The framework is demonstrated on 58 manufactured surfaces (10 materials, 10 processes), phase-extracted interferometric profilometry of a certified roundness standard, and 2D binary embeddings of prime numbers. A sparse-integer control proves the prime $β=2.15$ signal is genuinely arithmetic ($Δβ=+0.68$ over random-integer control), while a Cantor-embedding null result ($β=1.40$, TOST $p<0.01$) demonstrates that 2D exclusion is embedding-created rather than intrinsic. Density-thinning experiments establish that $β$ captures exclusion strength rather than point density, while absolute values are density-dependent. A distinct CSR baseline for binary fields at low fill fraction is identified, with a decision table provided. The $β$--$r_{\text{excl}}$ calibration, the CSR baseline correction, and the control protocols together constitute a calibrated measurement framework for reproducible characterisation of short-range exclusion in 2D spatial point processes.

We study spectral concentration bounds for kernel matrices as approximation of the corresponding kernel integral operator. Results are established under weak assumptions on the data setting and the reproducing kernel relying only on a Mercer condition and a local Weyl law. This allows us to deal with key features of kernel matrices, such as large multiplicities, large effective dimension, and heavy-tailed distributions. Our results apply to infinite dimensional principal component analysis, manifold learning, and Bayesian nonparametric statistics. We illustrate this via two prototypical examples: The heat kernel on the sphere and a wavelet prior from Bayesian nonparametrics.

In this paper, we consider MCMC algorithms for Student-$t$ regression models. We investigate the efficiency of Markov chains based on the algorithms in terms of whether trace-class results hold or not. We first consider the case where the parameters follow a matrix-normal-inverse-Wishart distribution and show that the Markov operator associated with a standard data augmentation algorithm is trace-class. We next consider the case of an improper prior and univariate outcomes. In this case, the standard Markov operator is not trace-class but the Markov operator associated with a collpased Gibbs algorithm is trace-class. Finally, we consider the case of an improper prior and multivariate outcomes. We obtain a trace-class result for a parameter expanded data augmentation algorithm which is based on a univariate working parameter.

This paper studies the problem of detecting a change in the distribution of a sequence of independent observations when the post-change distribution is unknown. We propose a novel change detection algorithm, termed Predictive-Mixture CuSum (PM-CuSum), which combines predictive distributions constructed from sliding windows of different lengths within a CuSum recursion. The predictive distributions are aggregated using adaptive weights based on their recent predictive performance. We show that PM-CuSum achieves first-order asymptotic optimality under mild conditions, and that its asymptotic delay bound has a smaller remainder order than what is achieved procedures using a single fixed (even oracle) window. Numerical simulations demonstrate that PM-CuSum performs well compared to existing methods. Moreover, it is demonstrated that forming likelihood ratios using full predictive distributions can substantially improve performance compared to plug-in likelihoods.

The emerging literature on hypothesis testing with data-dependent and post-hoc significance levels relies on a particular extension of the Type-I error to data-dependent levels. Existing arguments for this extension are heuristic, and primarily motivated by a resulting connection to the E-value. Our main contribution is to argue that the extension is 'right', by showing that it emerges from three axioms: within a large class of possible extensions, it is the only extension that nests classical Type-I error validity for data-independent levels, preserves classical validity for data-dependent levels and is monotone in the strength of the rejection claim. As a second contribution, we apply this result to support the common definition of the E-value, by showing that it arises as the 'right' notion of validity for the numerical representation of a generalized hypothesis test that may reject at different data-driven significance levels.

Large language models now serve millions of users daily, with providers incurring costs exceeding $700,000 per day. Each request requires token-by-token inference, making GPU scheduling central to latency, capacity, and cost. The difficulty is endogenous memory growth: generated tokens expand the Key-Value (KV) cache, and overflow can evict in-progress requests and waste prior computation. We formulate inference as a multi-stage online scheduling problem with endogenous memory growth, linear iteration times, and GPU-resident KV-cache constraints. We introduce a fluid model that characterizes equilibrium batch composition, memory requirement, and stability region. Guided by the fluid model, we design WAIT (Waiting for Accumulated Inference Threshold), a threshold-based admission rule for known output lengths, and Nested WAIT, which extends the rule to unknown output lengths by regulating how requests advance across decode-stage segments. Both algorithms approximate the fluid benchmark asymptotically under the stated memory conditions. Nested WAIT uses an additional safety buffer of moderate scale to hedge against memory-overflow-induced evictions under unknown output lengths. In Vidur simulations configured for Llama-2-7B on an A100 GPU, with supplemental real-GPU validation reported in the appendix, the policies enlarge the empirically observed stable operating range relative to widely used baseline algorithms and reduce latency especially in near-overloaded and overloaded regimes.

We develop a theoretical foundation for robust model comparison in a class of non-ergodic continuous volatility regression models contaminated by finite-activity jumps. Using the density-power weighting and the Hölder(-inequality)-based normalization of the conventional Gaussian quasi-likelihood function, we propose two Schwarz-type statistics and also establish their model selection consistency with respect to the minimal true parametric volatility coefficient. Numerical experiments are conducted to illustrate our theoretical findings.

Under network interference, a unit's observed outcome depends on the treatment assignment of its neighboring units in an exposure graph. Existing design-based asymptotic theory typically considers local interference by restricting neighborhood sizes in the exposure graph. Such methods do not apply to dense exposure graphs, so prior work has often adopted a superpopulation approach instead, imposing regularity through random-graph models. In this paper, we introduce a notion of convergence for a sequence of finite populations under anonymous interference. Building on the graph limit framework of Lovász and Szegedy, we show that large-scale geometry of the exposure graph can provide a source of regularity beyond sparsity assumptions or random-graph modeling. Under Bernoulli assignment, our convergence notion yields asymptotic normality of standard estimators for the average direct effect, even on dense, non-random exposure graphs. As a special case, graphon-based random-graph models studied in prior work generate finite populations that converge in our sense. Under these models, graph randomness generates exposure graphs with stable large-scale geometry, while first-order uncertainty in average direct effect estimation is driven by treatment assignment.

Membership Inference Attacks (MIAs) aim to distinguish training points (members) from unseen data (non-members), and are widely used to quantify memorization and assess privacy risks. Standard MIA evaluation requires repeated retraining, which is computationally costly for large models. One-run (single training with randomized data inclusion) and zero-run (post hoc evaluation) methods are often used instead, but their statistical validity remains unclear. We address this gap by framing MIA evaluation as a causal inference problem, defining \emph{memorization as the causal effect of including a data point in the training set}. This novel formulation reveals and formalizes key sources of bias in existing protocols: one-run methods suffer from interference between jointly included points, while zero-run evaluations are additionally confounded by distribution shift between member and non-member evaluation data. We derive causal analogues of standard MIA metrics and propose practical estimators for multi-run, one-run, and zero-run regimes with non-asymptotic consistency guarantees. We validate our approach in several settings, including pretrained and fine-tuned LLMs, showing that it enables reliable measurement of MIA performance without retraining and under distribution shift. Overall, our framework provides a principled foundation for privacy evaluation in modern AI systems.

Despite numerous attempts at mitigation since the inception of language models, hallucinations remain a persistent problem even in today's frontier LLMs. Why is this? We review existing definitions of hallucination and fold them into a single, unified definition wherein prior definitions are subsumed. We argue that hallucination can be unified by defining it as simply inaccurate (internal) world modeling, in a form where it is observable to the user. For example, stating a fact which contradicts a knowledge base OR producing a summary which contradicts the source. By varying the reference world model and conflict policy, our framework unifies prior definitions. We argue that this unified view is useful because it forces evaluations to clarify their assumed reference "world", distinguishes true hallucinations from planning or reward errors, and provides a common language for comparison across benchmarks and discussion of mitigation strategies. Building on this definition, we also connect our framework to HalluWorld, a complementary benchmark that instantiates fully specified reference world models for stress-testing model hallucinations.

This work presents the first systematic development of Stein's method for matrix distributions. We establish the basic essential ingredients of Stein's method for matrix normal approximation: we derive an extended-generator-based Stein identity from a matrix Ornstein-Uhlenbeck diffusion with two-sided scales, provide an explicit semigroup representation for the solution of the Stein equation, and obtain regularity estimates for the solution. The new methodology is demonstrated in three examples: (i) smooth Wasserstein distance bounds to quantify the matrix central limit theorem (a didactic example), (ii) a Wasserstein distance bound for the matrix normal approximation of the centered matrix $T$ distribution, and (iii) a Stein's method-of-moments approach to estimating the row and column covariance factors of the matrix normal, yielding a flexible class of weighted flip-flop Stein estimators that generalize Dutilleul's classical flip-flop algorithm and naturally accommodate row/column importance weights, systematic missingness, and projection onto structured covariance families. The latter two examples are intrinsically matrix-valued and cannot be treated using naive vectorization.

We consider statistical inference for a class of continuous semimartingale regression models based on high-frequency observations subject to contamination by finite-activity jumps and spike noise. By employing density-power weighting and Hölder-inequality-based normalization, we propose easy-to-implement, robustified versions of the conventional Gaussian quasi-maximum-likelihood estimator that require only a single tuning parameter. We prove their asymptotic mixed normality at the standard rate of $\sqrt{n}$. It is theoretically shown that these estimators are simultaneously robust against contamination in both the covariate and response processes. Additionally, under suitable conditions on the selection of the tuning parameter, the proposed estimators achieve the same asymptotic distribution as the conventional estimator in the contamination-free case. Illustrative simulation results highlight the estimators' insensitivity to the choice of the tuning parameter.

We propose an extreme dimension reduction method extending the Extreme-PLS approach to the discretized functional framework, where the covariate lies in the infinite-dimensional Hilbert space $L^2([0,1])$ but is partially observed on a dense time grid. The ideas are partly borrowed from both Partial Least-Squares (PLS) and Sliced Inverse Regression (SIR) techniques. Accordingly, the method relies on the projection of the covariate onto a subspace and maximizes the covariance between its projection and the response conditionally on an extreme event capturing the tail-information. The covariate and the heavy-tailed response are supposed to be linked through a non-linear inverse single-index model and our goal is to infer the index in this regression framework. We propose a new family of estimators and show its asymptotic consistency with convergence rates under the model. Assuming mild conditions on the noise, most of the assumptions are stated in terms of regular variation unlike the standard literature on SIR and single-index regression. In addition, we expand the theoretical analysis with a model-free almost sure consistency result for the empirical tail-moments in a general separable Hilbert space. Finally, our results are illustrated on a finite-sample study with synthetic functional data as well as high-frequency financial data, highlighting the effectiveness of the dimension reduction for capturing tail dependence and for extreme risk management.

Editing the content of an image with a pretrained text-to-image model remains challenging. Existing methods often distort fine details or introduce unintended artifacts. We propose using \emph{coupled stochastic differential equations} (coupled SDEs) to guide the sampling process of any pre-trained generative model that can be sampled by solving an SDE, including diffusion and rectified flow models. By driving both the source image and the edited image with the same correlated noise, our approach steers new samples toward the desired semantics while preserving visual similarity to the source. The method works out-of-the-box, without retraining or auxiliary networks, and achieves high prompt fidelity along with near-pixel-level consistency. These results position coupled SDEs as a simple yet powerful tool for controlled generative AI. Project page: https://z-jianxin.github.io/syncSDE-release/. Code: https://github.com/Z-Jianxin/syncSDE-release.

We consider the problem of estimating a time-varying sparse precision matrix, which is assumed to evolve in a piecewise constant manner. Building upon the Group Fused LASSO and LASSO penalty functions, we estimate both the precision matrix and the change points. We propose an alternative estimator to the commonly employed Gaussian likelihood loss, namely the D-trace loss. We provide the conditions for the consistency of the estimated change points and of the sparse estimators in each block. We show that the solutions to the corresponding estimation problem exist when some conditions relating to the tuning parameters of the penalty functions are satisfied. Unfortunately, these conditions are not verifiable in general, posing challenges for tuning the parameters in practice. To address this issue, we introduce a modified regularizer and develop a revised problem that always admits solutions: these solutions can be used for detecting possible unsolvability of the original problem or obtaining a solution of the original problem otherwise. An alternating direction method of multipliers (ADMM) is then proposed to solve the revised problem. The relevance of the method is illustrated through numerical experiments.

Large Language Models (LLMs) exhibit potentially harmful biases that reinforce culturally embedded stereotypes, influence moral judgments, or amplify positive evaluations of majority groups. We propose metacognitive myopia as a cognitive-ecological framework accounting for a conglomerate of established and emerging LLM biases. Our theoretical framework posits that biased samples in the information environment cause five symptoms of metacognitive myopia in LLMs: integration of invalid embeddings, susceptibility to redundant information, neglect of base rates in conditional computation, decision rules based on frequency, and inappropriate higher-order statistical inference for nested data structures. Moreover, it posits that the two main components of metacognition, monitoring and control, could account for these five symptoms. Accordingly, we further outline how monitoring and control could be approximated technically, for instance, through hidden parallel reasoning histories that allow interactive LLMs to evaluate risks of myopic inference before generating overt responses. Our theoretical framework provides a novel perspective on flawed human-machine interactions and agentic AI and raises significant ethical concerns regarding the implementation of LLMs in organizational structures and high-stakes decisions.

Bayesian optimization is highly effective for optimizing expensive-to-evaluate black-box functions, but it faces significant computational challenges due to the cubic per-iteration cost of Gaussian processes, which results in a total time complexity that is quartic with respect to the number of iterations. To address this limitation, we propose a novel algorithm, Bayesian optimization by kernel regression and density-based exploration (BOKE). BOKE uses kernel regression for efficient function approximation, kernel density for exploration, and integrates them into the confidence bound criteria to guide the optimization process, thus reducing computational costs to quadratic. Our theoretical analysis rigorously establishes the global convergence of BOKE under noisy evaluations. Through extensive numerical experiments on both synthetic and real-world optimization tasks, we demonstrate that BOKE not only performs competitively compared to Gaussian process-based methods and several other baseline methods but also exhibits superior computational efficiency. These results highlight BOKE's effectiveness in resource-constrained environments, providing a practical approach for optimization problems in engineering applications.

We address two important statistical problems: that of estimating mixtures of multivariate normal distributions and mixtures of $t$-distributions based on univariate projections, and that of quantifying a discrepancy between mixture distributions induced by two model-based clusterings. In the second problem, rather than introducing a direct metric on partitions, we propose a model-based distributional discrepancy between the fitted mixture distributions associated with two clusterings. The results are based on an earlier work of the authors, where it was shown that mixtures of multivariate Gaussian or $t$-distributions can be distinguished by projecting them onto a certain predetermined finite set of lines, the number of lines depending only on the total number of distributions involved and on the ambient dimension. We also compare our proposal with robust versions of the expectation-maximization method EM. In each case, we present algorithms for effecting the task, and compare them with existing methods by carrying out some simulations.

Propensity score (PS) methods are widely used to estimate treatment effects in non-randomized studies. Variance is typically estimated using sandwich or bootstrap methods, which can either treat the PS as estimated or fixed. The latter is thought to be conservative. Comparisons between the sandwich and bootstrap estimators have been compared in moderate to large sample sizes, favoring the bootstrap estimator. With the growing interest in treatments for rare disease and externally controlled clinical trials, very small sample sizes are not uncommon and the asymptotic properties of sandwich estimators may not hold. Bootstrap methods that allow for PS re-estimation can also generate problems with quasi-separation in small samples. It is unclear whether it is safe to prefer sandwich estimators or to assume that treating the PS as fixed is conservative. We conducted a Monte Carlo simulation to compare the performance of bootstrap versus sandwich variance and CI estimators for average treatment effects estimated with PS methods. We systematically evaluated the impact of treating the PS as fixed versus re-estimating it. These methodological comparisons were performed using Inverse Probability of Treatment Weighting (IPTW) and Augmented Inverse Probability of Treatment Weighting (AIPW) estimators. Simulations assessed performance under various conditions, including small sample sizes and different outcome and treatment prevalences. We illustrate the differences in our motivating example, the LIMIT-JIA trial. We show that the sandwich estimators can perform quite poorly in small samples, and fixed PS methods are not necessarily conservative. A stratified bootstrap avoids quasi-separation and performs well. Differences were large enough to alter statistical conclusions in our motivating example, LIMIT-JIA.

The growing number of infectious disease outbreaks, like the one caused by the SARS-CoV-2 virus, underscores the necessity of actuarial models that can adapt to epidemic-driven risks. Traditional life insurance frameworks often rely on static mortality assumptions that fail to capture the temporal and behavioral complexity of disease transmission. In this paper, we propose an integrated actuarial framework based on the SEIARD epidemiological model. This framework enables the explicit modeling of incubation periods and disease-induced mortality. We derive key actuarial quantities, including the present value of annuity benefits, payment streams, and net premiums, based on SEIARD dynamics. We formulate a prospective reserve function and analyze its evolution throughout the course of an epidemic. Additionally, we examine the forces of infection, mortality, and removal to assess their impact on epidemic-adjusted survival probabilities. Numerical simulations implemented via a nonstandard finite difference (NSFD) scheme illustrate the model's applicability under various parameter settings and insurance policy assumptions.

In a world awash with data, the ability to think and compute with data has become an important skill for students in many fields. For that reason, inclusion of some level of statistical computing in many introductory-level courses has grown more common in recent years. Existing literature has documented multiple success stories of teaching statistics with R, bolstered by the capabilities of R Markdown. In this article, we present an in-class data visualization activity intended to expose students to R and R Markdown during the first week of an introductory statistics class. The activity begins with a brief lecture on exploratory data analysis in R. Students are then placed in small groups tasked with exploring a new dataset to produce three visualizations that describe particular insights that are not immediately obvious from the data. Upon completion, students will have produced a series of univariate and multivariate visualizations on a real dataset and practiced describing them.