In recent years, there has been much progress toward the development of methods for converting three- and five-number summary statistics (i.e. minimum, maximum, median, and quartiles) to means and standard deviations (SDs). This is commonly done in the meta-analysis setting, where some studies report means and SDs, while other report quantile summaries. However, we show that three-number summaries, which are the most common, do not contain enough information to reliably estimate SDs. We show that very poor estimates can result, which may invalidate any inference and provide details of a sensitivity analysis that can allow researchers to have greater confidence in their results, or highlight potential sources of bias. We further explore whether nominating additional information can provide enough information regarding the unknown data shape to improve SD estimations, and in doing so introduce a new estimator using the scaled Beta distribution. Simulations and a real data example are used to highlight the advantages and disadvantages of this approach. A Web application is also provided to help researchers perform sensitivity analyses.

In survival analysis, dependent censoring poses significant challenges in accurately estimating model parameters and survival functions. This study introduces a novel framework leveraging Extended Generalized Marshall-Olkin (EGMO) models to address dependent censoring mechanisms. Geometric optimization techniques are employed to develop efficient estimation procedures that capture dependencies between failure and censoring times. We establish their asymptotic properties. Simulation studies and real data applications illustrate the method's robustness and effectiveness.

We study finite-horizon queue peaks in generalized switches, a standard stochastic-network model in which many queues share constrained service resources. Arrivals may be dependent, time-varying, and adapted to the past; the standing load condition is uniform interior slack, meaning the conditional mean arrival vector stays in a fixed contraction of the capacity region. We show that this slack reshapes the finite-time peak law for drift-minimizing scheduling policies such as MaxWeight. The square-root envelope that is sharp without slack persists only up to a geometry-dependent threshold; beyond that threshold, the running maximum grows only logarithmically with the horizon, both with high probability and in expectation. The mechanism is self-normalization: in the current queue direction, the projected fluctuation scale is normalized by the stabilizing drift scale. This removes capacity geometry from the logarithmic coefficient, while geometry remains in the threshold. Matching lower bounds show that both the logarithmic term and a geometric threshold are unavoidable. When finite-time state-space collapse is available, the threshold can be sharpened using local bottleneck geometry. For generalized input-queued switches, we obtain finite-time peak bounds with tight logarithmic coefficients. Simulations illustrate the two-phase envelope, local geometric refinements, and variance-sensitive improvements predicted by the theory.

This paper extends classical conformal frameworks for constructing prediction intervals with global marginal coverage $1-\alpha$ to intervals that provide explicitly calibrated guarantees for the upper and lower tails separately. Focusing on split conformal prediction, we first construct lower and upper one-sided conformal intervals that achieve marginal validity, and then derive the induced two-sided interval by intersection. Theoretical results prove both tail-specific and global marginal coverage of the induced two-sided interval. Results are presented first for the exchangeable setting, where coverage has finite-sample guarantees, and then for non-exchangeable data, where guarantees are asymptotic. Simulation studies show that the proposed approach achieves improved directional calibration relative to classical two-sided intervals, especially relevant in skewed data. Finally, the benefit of the proposed framework is showcased in a financial application, where one aims for return maximization while seeking strict control on the left tail.

Kunchenko's method of polynomial maximization provides a semiparametric apparatus for parameter estimation under non-Gaussian errors, but its classical power basis relies on finite higher-order integer moments. This paper introduces the Parametrically Adaptive Transition Polynomial (PATP), a signed-parity fractional-power family controlled by a continuous parameter alpha in [0,1]. The quadratic exponent map p_i(alpha) connects the fractal regime p_i(0)=1/i, the degenerate linear point p_i(1/2)=1, and the signed-parity integer-power regime p_i(1)=i. For the degree-S=2 case we derive a closed-form variance-reduction coefficient g_2(alpha) in terms of signed and absolute fractional moments, identify the singular behavior at alpha=1/2, and state the moment and regularity conditions under which the formula is meaningful. The construction should be read as a Form-B PATP analogue within Kunchenko's generalized apparatus, not as an exact recovery of the canonical even-power PMM basis at alpha=1. Numerical illustrations on canonical distributions are used to examine the finite-sample behavior of the signed-parity estimator and to mark the boundary of applicability for extremely heavy-tailed cases such as Cauchy.

Local Polynomial Regression (LPR) is a widely used nonparametric method for modeling complex relationships due to its flexibility and simplicity. It estimates a regression function by fitting low-degree polynomials to localized subsets of the data, weighted by proximity. However, traditional LPR is sensitive to outliers and high-leverage points, which can significantly affect estimation accuracy. This paper revisits the kernel function used to compute regression weights and proposes a novel framework that incorporates both predictor and response variables in the weighting mechanism. The focus of this work is a conditional density kernel that robustly estimates weights by mitigating the influence of outliers through localized density estimation. The proposed method is implemented in Python and is publicly available at this https URL. The population analysis quantifies the bias induced by density-based robust weighting, and the reported experiments show lower empirical bias than iterative robust LOWESS while remaining competitive with standard LOWESS. This advancement provides a promising extension to traditional LPR, opening new possibilities for robust regression applications.

Differential item functioning (DIF) detection is an important yet understudied problem in computerized adaptive testing (CAT). In this article, we proposed a two-level logistic model to improve DIF detection in CAT by explicitly accounting for nuisance effects arising from CAT-induced structural dependency. First, we conceptualized that adaptive item selection induces systematic dependencies among examinees and items through provisional ability estimates, whereas traditional single-level DIF methods assume independent observations and may yield misleading results in CAT settings. Then, using a numeric example and Monte Carlo simulations, we compared our proposed two-level model with competing single-level models under various CAT conditions, manipulating test length, exposure control, ability estimator, DIF type, and DIF prevalence. Item-level Type-I error and statistical power conditional on joint model convergence were reported for each model. We showed that the proposed two-level model has improved control of spurious DIF and competitive power relative to single-level models, particularly with shorter tests and smaller exposure rates. However, we observed that the model convergence varied systematically across simulated conditions, highlighting that inferential accuracy and convergence reliability are intertwined in complex CAT DIF settings. Through this study, we underscored both the promise of multilevel DIF modeling in CAT and the need for future research to jointly evaluate convergence and inferential performance when assessing DIF models.

Categorical variables are of uttermost importance in biomedical research. When two of them are considered, it is often the case that one wants to test whether or not they are statistically dependent. We show weaknesses of classical methods -- such as Pearson's and the G-test -- and we propose testing strategies based on distances that lack those drawbacks. We first develop this theory for classical two-dimensional contingency tables, within the context of distance covariance, an association measure that characterizes general statistical independence of two variables. We then apply the same fundamental ideas to one-dimensional tables, namely to the testing for goodness of fit to a discrete distribution, for which we resort to an analogous statistic called energy distance. We prove that our methodology has desirable theoretical properties, and we show that we can calibrate the null distribution of our test statistics without resampling. We illustrate all this in simulations, as well as with some real data examples, demonstrating the adequate performance of our approach for biostatistical practice.

Binary data factorization is common, but real-valued methods ignore discreteness and yield hard-to-interpret factors. Boolean Matrix Factorization (BooMF) instead decomposes a binary matrix into two lower-rank binary matrices via logical AND and OR, expressing the data as a Boolean disjunction of interpretable patterns. In cancer genomics, BooMF can reveal coordinated feature changes that may drive tumor evolution, unlike rotational or additive decompositions. Most existing BooMF methods are heuristic, greedy, sensitive to initialization, prone to local optima, and do not support principled model selection or uncertainty quantification. We introduce Bayesian Boolean Matrix Factorization (BBMF), a fully conjugate generative model with sparsity-inducing priors. It enforces Boolean constraints, yields interpretable latent factors with coherent uncertainty quantification, and admits Gibbs sampling with closed-form full conditionals. Because cancer evolution often involves widespread, near-simultaneous chromosome-number changes (e.g., whole-genome duplication followed by instability and selection), Boolean factorizations capture these patterns more naturally than additive models. Applied to arm-level copy-number alteration data in multiple myeloma, where entries indicate presence/absence of chromosomal-arm amplifications, BBMF finds a small set of interpretable bicliques linking patient subsets to recurrently co-altered chromosomal arms, providing a compact, biologically meaningful summary of tumor heterogeneity and demonstrating BBMF's utility for uncovering discrete latent structure in complex binary data.

Space-based monitoring of atmospheric carbon dioxide (CO2) is essential for constraining the global carbon budget. NASA's Orbiting Carbon Observatory-2 (OCO-2) estimates column-averaged dry-air mole fractions of CO2 (XCO2) using high-resolution spectra. However, current operational retrieval algorithms are computationally expensive and do not properly quantify uncertainties. We present a novel deep learning framework that addresses these challenges. Due to the difficulties of ground-truth data for real satellite observations, we develop and validate our approach using a high-fidelity simulation dataset. This dataset, created to support OCO-2 uncertainty quantification (UQ), incorporates realistic forward model errors. Our architecture encodes spectral bands using a multi-branch neural network and estimates posteriors of the full CO2 column or desired summaries thereof using two scalable UQ methods: Laplace approximations and normalizing flows. Our approach has five key advantages relative to operational "full-physics" solvers: (1) Amortization: Inference is orders of magnitude faster, enabling real-time processing of massive data streams; (2) Model error robustness: By training on simulations that explicitly include model discrepancies, our method accounts for systematic errors often neglected by standard inversions; (3) Point estimate accuracy: We achieve superior predictive accuracy compared to baseline methods; (4) Improved UQ: The probabilistic outputs yield better-calibrated uncertainty estimates; and (5) Non-Gaussian posteriors: When utilizing normalizing flows, our framework successfully models complex, asymmetric posterior distributions, overcoming the limitations of the Gaussian assumption. These results suggest that simulation-based deep learning is a viable path toward next-generation operational processing systems.

We propose a novel framework for uncertainty quantification using compressed sensing magnetic resonance image reconstruction. The problem is formulated within a Bayesian framework as a linear inverse problem, with prior distributions assigned to the unknown model parameters. Specifically, the image to be reconstructed is assumed to be sparse in a given basis. We develop a general framework applicable to any basis and as examples, we test the sparsity of the image in its (1) spatial gradients using a total variation prior model, and in its (2) wavelet transform. A Markov chain Monte Carlo (MCMC) method, based on a split-and-augmented Gibbs sampler, is then employed to sample from the posterior distribution of the unknown parameters. The non-differentiable conditional distributions are efficiently sampled using a proximal MCMC method. The proposed algorithms are validated on both single-coil and multi-coil datasets using various k-space sub-sampling patterns and ratios. The results demonstrate the superior performance of each proposed approach in reconstructing images compared to its counterpart optimisation-based method. Moreover, our framework effectively quantifies uncertainty, showing a notable correlation between estimated uncertainty maps and error maps computed using ground truth and reconstructed images, compared with existing deep learning-based methods.

We study sparse semi-continuous tensor data with excess zeros, heavy right tails, and slice-specific dispersion. Such features arise naturally in monetary-valued multi-way data, such as international trade, where most exporter--importer--product--year cells are zero while positive values are continuous and highly variable. To model these data, we propose a Bayesian hierarchical tensor factorization model that places a low-rank CP structure on a latent Poisson rate tensor and couples it with a conditional Gamma model for positive outcomes, with rate parameters that can vary across slices within a mode. The model therefore separates the occurrence and magnitude of positive observations while borrowing strength across all tensor dimensions through a shared low-rank latent structure. To scale posterior inference to large arrays, we develop a hybrid variational--Monte Carlo algorithm that combines efficient coordinate ascent updates with a partially collapsed augmented-data sampler. Applied to approximately 60 million trade flows, the method surfaces multiway dependence across exporters, importers, products, and years that is difficult to recover from gravity-type or pairwise network analyses, which do not jointly model the product and temporal dimensions.

Spatial fields in the Earth and environmental sciences are often available at multiple scales or resolutions. While coarse-scale data (e.g., from global circulation models) are often abundant, they lack the local detail provided by fine-scale data (e.g., from regional climate models), which are typically computationally expensive to generate. Statistical downscaling and multi-scale data fusion address this challenge by predicting high-resolution fields from low-resolution or related inputs. We propose a highly scalable Bayesian approach that can learn the joint non-Gaussian distribution and nonlinear dependence structure of nonstationary spatial fields across multiple scales from a small number of training samples. Our method employs scale-aware autoregressive Gaussian processes with suitably chosen regularization-inducing priors to model the conditional distribution of fine-scale fields given coarse-scale data. Exploiting conjugacy, the integrated likelihood is available in closed form, enabling efficient parameter optimization via stochastic gradient descent. Once trained, the method provides a closed-form characterization of the posterior distribution of fine-scale fields given coarse-scale inputs. In numerical comparisons, we demonstrate that our approach substantially outperforms existing methods and effectively characterizes and simulates fine-scale climate behavior based on output from coarse global circulation models.

Online A/B tests are the standard tool for data-driven decision-making at scale. Among the design choices with the largest impact on statistical power is the triggering mechanism: how many users to expose and for how long. This often requires forecasting user engagement, i.e., whether enough users will trigger, and when a target participation level will be reached, from limited pilot data. We introduce a Bayesian nonparametric model for predicting both new-user counts and total triggers, accommodating the heavy-tailed engagement patterns typical of web experiments. All predictive quantities can be computed without intensive numerical procedures such as MCMC or variational inference. We evaluate on three public datasets (over 450 public benchmark evaluations) and 1,774 proprietary A/B tests. In all the settings, our models show improved accuracy in forecasting new users, total triggers, and time to reach a target sample size compared with state-ofthe-art competitors, especially when only a few pilot days are observed.

Feature allocation models are Bayesian nonparametric tools tailored to data in which each observation can simultaneously exhibit multiple characteristics, or features. A fundamental limitation of standard formulations is that feature labels are assumed to be independent and identically distributed, and therefore play no role in posterior inference. The present paper introduces a unified Bayesian framework for extended feature allocation models, in which feature labels and proportions are modeled jointly, thereby enabling the simultaneous discovery of features and learning of dependencies among their labels. Building on point process theory, we develop a full Bayesian analysis of these models. Within this general setting, we also characterize previously proposed priors as those leading to poor predictive distributions, which cannot capture label dependencies and are insensitive to the observed frequency spectrum. Our methodology is designed to move beyond such standard formulations by leveraging the information carried by feature labels. We demonstrate the usefulness of our approach by introducing: (i) a Cox process prior that clusters genomic variant embeddings while predicting new variants and new variant clusters; (ii) a determinantal point process prior for repeated forest surveys, where prediction concerns both the number and the locations of unobserved trees.

Consider an array receiving unknown wideband signals from an unknown number of sources $k$. Wideband signals can occupy arbitrarily wide bandwidths, rendering demodulation-based approaches inapplicable, a common situation in settings involving acoustic signals. Here, we aim to determine $k$ given $N$ noisy array-valued measurements, a task known as the "detection problem," for which Bayesian model comparison is a common approach. To render Bayesian inference tractable, it is typically necessary to marginalize the source signals. Unfortunately, for wideband signals, naive marginalization has an unaffordable time complexity of $\mathcal{O}(N^3 k^3)$. As a result, fully Bayesian signal detection has yet to be demonstrated in wideband settings. In this work, we propose a wideband signal model that allows for computationally tractable marginalization of the source signals. We begin from the canonical model of linear time-invariant (LTI) signal propagation, which is then augmented into a circular convolution, all without loss of generality. This allows for efficient computation in the frequency domain, where the resulting linear system admits a decomposition into a sparse matrix we refer to as a \textit{stripe matrix decomposition}. Exploiting this sparsity pattern reduces the time complexity of computing the marginal likelihood to $\mathcal{O}(N k^3)$. These computational improvements enable efficient posterior inference via reversible-jump Markov chain Monte Carlo (RJMCMC). In this work, we use the non-reversible extension of RJMCMC (NRJMCMC), which often achieves lower autocorrelation and faster convergence than RJMCMC. Detection of the latent source signals can then be performed in a fully Bayesian manner using samples drawn by NRJMCMC. We evaluate our procedure by comparing it against generalized likelihood ratio testing (GLRT) and information criteria.

Whether connected units are similar because influence spreads across ties or because similar units form ties, is a long-standing problem. Contagion or influence is generically unidentified from observational network data. We consider the minimal and common setting of a single network, fixed over time, with two waves of a binary nodal outcome. Rather than positing a parametric model for network formation, we reframe identification of contagion as a selection-bias problem and develop a sensitivity framework. We define a controlled direct effect (CDE) holding a tie present while intervening on an alter's outcome. We show that the gap between the CDE and the observed connected-dyad risk ratio is governed by how strongly a latent homophily variable shifts the composition of connected dyads. Inspired by Smith-style selection-bias sensitivity analysis and the risk-ratio bounding function of Ding and VanderWeele we develop interpretable nonparametric bounds. This translates the question "is there contagion?" into the question "how strong would latent homophily have to be to explain away the observed contagion?" A simulation study characterizes the bounds' error control and power. We apply the framework to the 2008 U.S. House votes on the Troubled Asset Relief Program, identifying under which assumptions contagion is plausible.

We develop an anytime-valid framework for optimal policy identification from logged contextual bandit data. In many applied settings, the analyst wants to select the optimal policy from a candidate policy class $\Pi$, but data are generated by an externally determined logging policy that they do not control. The analyst may also wish to monitor evidence continuously, stopping once the optimal policy is clear rather than committing to a fixed sample size in advance. This paper addresses these challenges by constructing a time-indexed set $S_t$ that retains the true optimal policy set uniformly over time with high probability. The resulting procedure allows the analyst to monitor policy values, eliminate clearly suboptimal policies, and stop at data-dependent times without invalidating inference. When the optimal policy is unique, we define a stopping time for its identification and derive a sample-complexity bound scaling as $O\!\left(\frac{\log |\Pi|+\log\log(1/\Delta_{\min})}{\Delta_{\min}^2}\right)$, where $\Delta_{\min}$ is the gap between the best and second-best policy values. Simulations demonstrate that the anytime-valid approach can yield substantial sample savings relative to fixed-$N$ designs. An application to a large adaptive experiment on reducing misinformation online illustrates how the method provides a dynamic view as evidence on the optimal policy accumulates.

We study causal effect estimation with compositional treatments, where the exposure lies on a simplex and the estimand is defined over compositions rather than scalar or binary values. By considering a projection of the average potential outcome onto the treatment space, a kernel-based covariate functional balancing approach is adopted for weight construction. The weights are obtained by directly minimizing a worst-case balancing error over a reproducing kernel Hilbert space (RKHS) defined on the joint space of treatments and covariates, instead of being estimated under a treatment assignment model. Building on these weights, an augmented weighted estimator (AWE) is proposed, where the outcome function is estimated via kernel ridge regression and combined with a marginal augmentation over the covariate distribution. Despite the complex structure of the resulting objective, a finite-dimensional convex optimization problem is formulated via a representer theorem and a low-rank approximation. The proposed estimator achieves $\sqrt{n}$-consistency without requiring consistent estimation or smoothness of the weights. An asymptotic normality result is established around a sample-specific target. Empirical performance is demonstrated through simulation studies and a real data application.

Mediation analysis is widely used to disentangle causal pathways, yet in many real-world studies the mediator M and outcome Y are never jointly observed. This incompleteness breaks the standard identification strategy for natural direct and indirect effects. We introduce a novel data fusion framework that restores the identification by combining two incomplete data sources, one measuring $M$ and the other measuring Y. Our approach leverages shared instrumental variables (IVs) to circumvent the need to observe (M,Y) jointly, remains valid under unmeasured confounding via a no-interaction condition, and accommodates covariate and exposure shifts across data sources under a latent alignment condition. We establish two identification strategies, one for settings with a known set of valid IVs, and another for settings where valid IVs must be learned. We further develop semiparametric, influence-function-based estimators with multiple robustness properties, and propose an estimator that attains the semiparametric efficiency bound under appropriate conditions. We apply our framework to quantify the extent to which the effect of SNP rs610932 on dementia risk is mediated through immune-related gene-expression pathways.

Causal discovery from observational data remains challenging due to the need to recover directed structure and latent confounding without interventions. We propose FoundCause, an amortized causal discovery model trained entirely on synthetic data that maps datasets directly to causal graphs in a single forward pass. By learning from large collections of simulated structural causal models, FoundCause captures transferable statistical patterns that generalize beyond individual datasets. The architecture incorporates several key inductive biases for causal discovery. It uses a permutation-invariant transformer encoder with alternating attention over samples and variables to jointly model cross-variable dependence and per-variable distributions. Pairwise statistical features derived from classical asymmetry measures are injected through statistics-conditioned attention, guiding the model toward known causal signals. A factorized decoder separates edge existence from direction, while a triangular refinement module enables reasoning over higher-order causal motifs such as chains and colliders. In addition, a dedicated confounder module based on learnable latent tokens explicitly models hidden common causes, and the model explicitly handles missing data via its masked input representation. To our knowledge, FoundCause is the first amortized causal discovery approach to explicitly model latent confounding. FoundCause outperforms 11 classical non-amortized methods (e.g., PC, GES, NOTEARS-style optimization) and 4 amortized causal discovery methods on 15 real-world datasets, achieving +9.6% improvement in $F_1$, +1.2% in AUROC, and an 18.9% reduction in structural Hamming distance relative to the strongest non-amortized methods, while performing inference in a single forward pass.

We develop a decision-theoretic framework for distributed Bayesian experimental design in which local agents evaluate candidate experiments using expected information gain and transmit their local design decisions to a fusion center. Unlike centralized Bayesian design, where all likelihood components and information-gain values are available to a single planner, the fusion center in the distributed setting chooses a global experiment from compressed local recommendations. We derive the Bayes-optimal fusion rule, which selects the experiment with largest conditional expected centralized information gain given the observed local design decisions. This rule is analogous in spirit to optimal fusion rules in distributed detection, but differs fundamentally because the underlying utility is expected information gain and the resulting loss is information-gain regret rather than classification error. We also establish information-loss bounds and identify conditions under which the decision-only fusion rule is asymptotically equivalent to the centralized design. Numerical experiments show that Bayes-optimal fusion closely approximates the centralized oracle, whereas majority voting can be highly suboptimal when a minority of sites carry disproportionate information.

Response-adaptive randomization (RAR) in clinical trials aims to improve ethical and statistical efficiency by dynamically allocating patients to treatments based on observed outcomes. While RAR based on a target optimal allocation have been extensively studied for two-arms settings, their extension to multi-treatment experiments ($K \geq 2$) remains theoretically fragmented, with most existing methods focusing on specific algorithms or restricted target allocations. In this paper, we introduce a unified framework for response-adaptive targeting, the $\alpha$-Rebalancing Targeting Strategies ($\alpha$RTS), which generalize the ERADE two-armed strategy of Hu et al. [2009]. We prove that all designs in this family share fundamental asymptotic properties: strong consistency, asymptotic normality of allocation proportions and treatment effect estimators, and asymptotic efficiency. To address sparse target regimes (where some treatments are asymptotically eliminated), we further propose $\alpha$RTS with Forced Exploration, a variant that guarantees infinite sampling for all treatments while preserving the asymptotic guarantees. Extensive simulations illustrate the finite-sample behavior of $\alpha$RTS variants in a 3-armed context, highlighting in particular the critical role of forced exploration in sparse settings.

Mediation analysis is essential for decomposing the causal effect of a treatment into direct and indirect pathways. However, many practical settings rely on the stringent assumption that recanting witnesses, defined as treatment-induced mediator-outcome confounders, are either absent or fully known a priori. Such a requirement is often untenable, especially when these variables remain unobservable due to measurement difficulties or privacy constraints. In this paper, we leverage proximal causal inference to develop three novel identification strategies to address the challenge of identifying path-specific effects in the presence of unknown recanting witnesses. Building on this, we develop a semiparametric inference framework that derives the efficient influence function and proposes a proximal multiply robust estimator, which remains consistent if at least one set of nuisance models is correctly specified. When all nuisance models are correctly specified and converge at appropriate rates, the estimator is asymptotically normal and achieves the semiparametric efficiency bound. We provide a minimax optimization-based debiased machine learning procedure for point estimation and constructing valid confidence intervals. The performance of the proposed methods is demonstrated by simulation studies and a real data application.

Organizations and researchers show increasing interest in using large language models (LLMs) in place of human participants in A/B tests, in the hope of experimenting faster and at lower cost. We study when a treatment effect estimated on LLM outcomes recovers the effect that would have been measured on the human population of interest. Distributional equivalence between LLM and human outcomes would make any standard estimator valid but is unrealistic. We therefore develop a statistical framework that adapts surrogate endpoint theory to LLMs. The framework shows that calibrating LLM outcomes to human outcomes identifies the average treatment effect under surrogacy and comparability conditions that are jointly weaker than distributional equivalence. When these conditions fail, the effect of interest is only partially identified, and we provide diagnostics that can falsify surrogacy on historical experiments together with a bound on the worst-case bias from limited overlap. We further show that the stochasticity inherent to LLMs introduces both bias and variance, but using an average of multiple draws as the surrogate mitigates both. We illustrate the methods and theory in simulations and an application to A/B tests on Upworthy headlines. A central takeaway from our work is that the validity of LLM outcomes as surrogates can only be falsified for past treatments and never verified for new ones, so human experiments remain indispensable for novel interventions. We discuss the role of LLM choice, prompting, and temperature as design variables, and how to size human experiments for validation.

Personalized medicine in acute ischemic stroke requires moving beyond average treatment effects (ATE) to individualized treatment effect (ITE) estimates to support treatment decisions. In acute ischemic stroke, mechanical thrombectomy has been shown to be more effective on average than lysis in randomized controlled trials (RCTs), such as the MR CLEAN study. We aim to identify which individual patients benefit most from mechanical thrombectomy compared to lysis. The outcome of interest is the modified Rankin Scale (mRS) at three months, an ordinal measure of functional disability (0: no symptoms, 6: death). We demonstrate that causal transformation models on directed acyclic graphs (TRAM-DAG) can be used for ITE estimation after being fitted on observational MAGIC multi-center stroke patient data. To ensure comparability with the MR CLEAN population, which we use for validation, we train the TRAM-DAG on a MAGIC sub-population with NIHSS at admission >= 6, corresponding to one inclusion criterion of MR CLEAN. The fitted model is then used to estimate ITEs for stroke patients in the MR CLEAN population. While these ITE estimates cannot be confirmed experimentally, we show that their average is consistent with the trial's reported ATE. Furthermore, the ITE estimates correctly rank trial patients by their observed frequency of a good outcome (mRS at three months <= 2). These findings support the use of causal models like TRAM-DAG for personalized decision-making in stroke care and highlight their ability to bridge the gap between observational evidence and clinical trials.

Missing data are ubiquitous in public health research. When estimating causal effects, there are well-established methods to address bias to due missing outcomes. Commonly, causal estimands are defined under hypothetical interventions to "set" the exposure and to prevent missingness. We demonstrate how this framework can be extended to missing exposures. We further extend this framework to incorporate missingness on the baseline outcome, which induces missingness on the population of interest (e.g., persons at-risk). To do so, we highlight Counterfactual Strata Effects, a general class of causal estimands where the focus population is subject to missingness and/or impacted by the exposure. They are termed such because the estimand involves conditioning on a counterfactual this http URL each setting, we present the causal model, relevant counterfactuals, causal estimand, and identification result. We demonstrate with a real-data example to investigate the effect of alcohol consumption on the risk of incident tuberculosis (TB) infection in rural Uganda. We highlight the use of TMLE with Super Learner for estimation and inference and discuss the practical consequences of our approach.

Instrumental variable (IV) and proximal causal learning (Proxy) methods are central frameworks for causal inference in the presence of unobserved confounding. Despite substantial methodological advances, existing approaches rarely provide reliable epistemic uncertainty (EU) quantification. We address this gap through a Deconditional Gaussian Process (DGP) framework for uncertainty-aware causal learning. Our formulation recovers popular kernel estimators as the posterior mean, ensuring predictive precision, while the posterior variance yields principled and well-calibrated EU. Moreover, the probabilistic structure enables systematic model selection via marginal log-likelihood optimization. Empirical results demonstrate strong predictive performance alongside informative EU quantification, evaluated via empirical coverage frequencies and decision-aware accuracy rejection curves. Together, our approach provides a unified, practical solution for causal inference under unobserved confounding with reliable uncertainty.

This study investigates supervised machine learning techniques for identifying ship hydrodynamic coefficients from CFD-generated data from free-running simulations. Specifically, ordinary least squares and regularized regression methods are applied to Abkowitz-type manoeuvring models. Training and validation datasets are derived from URANS simulations of zig-zag and turning circle manoeuvres, which are validated against experimental benchmark data. The analysis evaluates the effects of coefficient set size, minimum training length required for predictive model training, and manoeuvre combinations on model performance. Results demonstrate the suitability of large-angle zig-zag manoeuvres for hydrodynamic system identification, provided that multicollinearity is addressed through appropriate coefficient selection, regression models, or input data variability. Larger coefficient sets offer greater model flexibility for variable conditions but are more prone to multicollinearity. Regularized regression techniques effectively mitigate multicollinearity and notably enhance prediction accuracy, as does incorporating more diverse manoeuvring data. Among tested models, Ridge regression provided the best compromise between computational efficiency and prediction accuracy.

Measurement-constrained problems frequently arise in modern applications such as electronic health record studies. In such problems, despite the availability of large datasets, collecting labeled data can be highly costly or time-consuming, allowing only a small portion of the data to be labeled within a given budget. This raises a critical question: which data points are most beneficial to label given the budget constraint? We study this question in the context of estimating an optimal individualized threshold under a measurement-constrained M-estimation framework. In particular, our goal is to estimate a high-dimensional parameter $\theta$ in a linear threshold $\theta^TZ$ for a continuous variable $X$ such that the discrepancy between whether $X$ exceeds the threshold $\theta^TZ$ and a binary outcome $Y$ is minimized. In the measurement-constrained setting, we propose a novel $K$-step active subsampling algorithm to estimate $\theta$, which iteratively samples the most informative observations in the dataset and solves a regularized M-estimator. Our theoretical analysis reveals a sharp phase transition phenomenon with respect to $\beta$, the smoothness of the conditional density of $X$ given $Y$ and $Z$. Please see the paper for the full abstract.

Geostatistical spatial prediction for environmental processes is typically undertaken using Gaussian process models via Kriging, while machine learning (ML) algorithms are state-of-the-art for non-spatial prediction. An exciting recent fusion of these ideas imbibes traditional ML algorithms with the capacity to deal with spatial autocorrelation, leading to improved predictive performance. A range of approaches have been proposed, including fusion with Gaussian processes, observation-driven correlation structures, spatial basis functions and local geographical fitting. However, there has been no numerical comparison of their relative predictive performances, which is needed to advise environmental scientists on the optimal approach to use. This paper fills this knowledge gap, and focuses on random forests as the ML algorithm because they are more computationally and conceptually straightforward to implement than deep learning algorithms. The results from two studies are presented, the first being a controlled simulation experiment investigating whether any single approach is consistently superior across different spatial autocorrelation types. The second study focuses on the prediction of air pollution concentrations within a tuberculosis prevalence study in Blantyre, Malawi. The results show that whilst no single approach is universally superior, utilising spatial basis functions appears to perform consistently well across both the simulation and real data studies.

Understanding urban mobility patterns is crucial for designing efficient and sustainable transportation systems. Motivated by an application to the municipality of Padova and its surroundings, we propose a novel statistical framework for the analysis and clustering of mobility trajectories derived from telephonic data. We introduce a compositional representation of individual movements that integrates the uncertain device location with information on the surrounding road network, encoding at each time point the proportions of different road types compatible with the observed position. This formulation naturally accounts for measurement uncertainty and yields trajectories evolving in the simplex. To model these data, we develop a state-space framework for compositional time series that captures both the telephonic measurement error and the temporal dynamics of the latent mobility process. Building on this representation, we propose a model-based clustering approach based on mixtures of state-space models to identify groups of trajectories with similar evolution. This allows us to aggregate individual movements into interpretable mobility patterns at the population level. The results of the case study demonstrate the ability of the approach to uncover meaningful mobility behaviors, providing insights that are potentially relevant to policy makers.

Understanding how individual metro usage evolves over multi-year horizons is essential for transit planning and passenger retention. However, existing approaches typically characterize mobility patterns as static clusters or short-term variability, leaving the lifecycle dynamics of transit participation underexplored. This study proposes a state-based lifecycle modeling framework that integrates Hidden Semi-Markov Models (HSMM) with discrete-time survival analysis to characterize the evolution of individual metro mobility. The HSMM infers latent mobility states with explicit duration distributions and a transition matrix governing regime changes, while the survival component models exit and re-entry events via state-dependent hazard functions conditioned on mobility-state trajectories and behavioral history. Applied to four years of smart card data from the Shanghai metro system (2021-2024), the framework enables the identification of interpretable mobility states, the characterization of transition dynamics, and the quantification of state-dependent exit and re-entry processes. The analysis reveals five robust mobility states with a directional transition hierarchy centered on an occasional-usage gateway state, and fundamentally different temporal mechanisms governing disengagement and return: exit hazard is state-dependent but duration-independent, whereas re-entry hazard decays sharply with inactivity length. These findings provide a methodological foundation for lifecycle-oriented mobility analysis and practical guidance for transit operators to identify at-risk users and time retention interventions.

Explaining precipitation behavior at daily scale is important for fine scale understanding of the mechanisms driving precipitation. However, this effort is challenging because of the frequent incidence of zeros. The challenge is amplified by the acknowledged incidence of two types of zeros -- absence of precipitation as a dry event and absence of measured precipitation due to detection limits. In this work, we propose a multilevel spatio-temporal model which allows us to distinguish and explain the two types of zeros, as well as to model positive precipitation above the detection limit. The methodology combines a point mass at zero with probability modeled through a probit regression, a Gamma regression for latent positive precipitation amounts, and an observation mechanism subject to threshold-induced censoring. To capture spatial dependencies, Gaussian processes are employed in each regression model. Working within a Bayesian framework, we can obtain a rich range of inference with exact uncertainty. In particular, we provide model-based inference tools to compare and quantify differences between the true precipitation process and its observed counterpart across relevant characteristics. We apply our model to the analysis of daily spring observations at 70 sites over 15 years from the Ebro River Basin in northeastern Spain. Our findings indicate that the threshold strongly affects the occurrence of observed precipitation, especially in humid regions. While its impact on total accumulated amounts is small, it can exert a relevant effect on upper quantiles.

For datasets with unknown but stationary serial dependence, a robust long run variance estimator is essential to handle diverse scenarios. Spectral variance estimators are commonly used but tend to exhibit significant negative bias in the presence of positive correlation. To overcome this, zero lugsail estimators have been introduced, offering zero asymptotic bias regardless of the correlation structure. However, there are currently no guidelines for selecting the optimal bandwidth for lugsail estimators, a critical component in the estimation process. We propose an inference optimal bandwidth rule for lugsail estimators, based on nonstandard fixed-smoothing limiting distributions developed in our study. This approach significantly improves bias correction, accounts for variability, and provides an estimator optimized for robust inference. Our theoretical findings are supported by a simulation study.

New York City implemented the nation's first cordon-based congestion pricing program in January 2025, providing an opportunity to evaluate how system-wide urban mobility responds to large-scale pricing interventions. Because such policies generate spillovers across modes and locations, credible control groups are difficult to construct. We address this challenge using time series foundation models to generate probabilistic counterfactual demand forecasts with calibrated uncertainty. Applying this framework to bus, subway, and aggregate trip volume data, we find that post-policy bus and subway ridership increased significantly relative to expected no-policy demand, while overall travel demand decreased modestly. The effects are spatially heterogeneous: while reductions in overall travel demand are concentrated within the Congestion Relief Zone, transit gains extend beyond Manhattan's core. Socio-demographic analyses further reveal uneven adaptation across neighborhoods, highlighting spatial equity implications. Our framework provides a scalable approach for the uncertainty-aware evaluation of system-wide urban interventions when clean control groups are unavailable.

We present a one-month-ahead conditional probabilistic framework for wind-power forecasting at ten-minute resolution. Monthly Weibull shape and scale parameters are estimated from serially dependent SCADA wind-speed data, corrected through a Godambe covariance, and forecast by a heteroskedastic Kalman filter on a bivariate VAR(1) state-space model. Conditional on the MMSE forecasted Weibull invariant law, we construct and compare three positive wind-speed SDE models: an Ornstein-Uhlenbeck-Weibull transform, a Fokker-Planck drift-first diffusion, and a Fokker-Planck diffusion-first model. The simulated wind-speed ensembles are mapped to power through a calibrated XGBoost power curve. Applied to January 2021 data from a Senvion MM92 turbine at Kelmarsh Wind Farm, the three SDE formulations are statistically indistinguishable in probabilistic accuracy, with mean CRPS values between 1.569 and 1.575 m/s. The diffusion-first model is therefore preferred on computational grounds, reducing runtime by about a factor of seven relative to the OU-Weibull model. In the power domain, the Wasserstein distance between simulated and observed distributions is 26.1-27.6 kW, below $1.4\%$ of rated capacity, while the monthly energy-yield bias is about $-7.3\%$ for the examined month. Exceedance-probability errors remain below 1.6 percentage points over the 0-1500 kW range and about 2.2 percentage points near rated power. These quantities provide decision-relevant probabilistic inputs for downstream operational problems, rather than completed reserve, storage, market, or fatigue-optimization decisions. Full marginalisation over the Kalman predictive law of the Weibull parameters is left as a natural extension.

In many insurance contexts, dependence between risks of a portfolio may arise from their frequencies. We investigate a dependent risk model in which we assume the vector of count variables to be a tree-structured Markov random field with Poisson marginals. The tree structure translates into a wide variety of dependence schemes. We study the global risk of the portfolio and the risk allocation to all its constituents. We provide asymptotic results for portfolios defined on infinitely growing trees. To illustrate its flexibility and computational scalability to higher dimensions, we calibrate the risk model on real-world extreme rainfall data and perform a risk analysis.

Assessing climate-driven mortality risk has become an emerging area of research in recent decades. In this paper, we propose a novel approach to explicitly incorporate climate-driven effects into both single- and multi-population stochastic mortality models. The new model consists of two components: a stochastic mortality model, and a distributed lag non-linear model (DLNM). The stochastic component captures the non-climate long-term trend, volatility, and seasonal patterns in mortality rates. The DLNM component captures non-linear and lagged effects of climate variables on mortality, as well as the impact of heat waves and cold waves across different age groups. For model calibration, we propose a novel backfitting algorithm that allows us to disentangle the climate-driven mortality risk from the non-climate-driven stochastic mortality risk. We illustrate the effectiveness and improved short-term (1--18 month) forecasting performance of our model against four alternative models, using data from three European regions: Athens, Lisbon, and Rome. Furthermore, as an application of the proposed modeling framework, we utilize future UTCI data generated from climate models to provide total mortality forecasts into 2045 across these regions under two Representative Concentration Pathway (RCP) scenarios, taking both stochastic mortality improvement trend and climate risk into account. The projections show a noticeable decrease in winter mortality alongside a rise in summer mortality, driven by a general increase in UTCI over time. Although we expect slightly lower overall mortality in the short term under RCP8.5 compared to RCP2.6, a long-term increase in total mortality is anticipated under the RCP8.5 scenario.

Small sample sizes pose significant challenges in regression analysis, often leading to violations of classical assumptions such as normality, homoscedasticity, and independence of residuals. These violations compromise parameter estimation accuracy, reduce statistical power, and limit the generalizability of findings. This study introduces the Gaussian Process-based Modified Extreme Value Theorem (GP-MEVT) method, a novel hybrid data augmentation approach that combines Gaussian Process with Extreme Value Theory to address these limitations. The GP-MEVT method generates augmented observations that extend the predictor space beyond the observed range while preserving the underlying linear structure and introducing controlled variability based on residual variation, through comprehensive simulation studies across three variance scenarios (sigma = 2, 5, 8) and sample sizes (n = 10, 15, 20). Here, we demonstrate that GP-MEVT achieves a higher rate of assumption satisfaction, substantially outperforming standard bootstrap and bootstrap with noise methods. The proposed method also exhibits reasonable parameter estimation accuracy, with intercept and slope estimates consistently closer to true parameter values, and maintains competitive or superior model fitting performance as measured by root mean square error. Application to a real-world dataset confirms these advantages, with GP-MEVT achieving a 67.1% assumption satisfaction rate compared to 17.3% and 21.2% for bootstrap alternatives. These findings establish GP-MEVT as a robust and reliable framework for fitting linear regression models to small datasets, offering practitioners a principled approach to statistical inference when sample size limitations are unavoidable.

Hidden Markov models are widely used to infer latent state sequences from sequential data, but Viterbi decoding reports only one most likely complete path. When decoded states carry scientific meaning, this single maximizer can conceal pathwise uncertainty created by multiple near-optimal trajectories. Conditional on a fitted HMM, we introduce the tropical Viterbi tube: the set of hidden trajectories whose complete-data log-score lies within a tolerance of the Viterbi optimum. State, transition, and change-status projections show which local features remain compatible with globally near-optimal complete paths, giving a pathwise uncertainty layer for HMMs in sequence analysis, ecology, finance, biomedical monitoring, and related domains. The tube is a posterior superlevel set on complete hidden-path space, with tolerance interpreted as a log posterior-odds loss relative to a Viterbi path. Calibrating the tolerance to a target posterior mass gives an HPD-threshold credible region for the complete latent path and conservative simultaneous projected bands. We prove monotonicity, step-function behavior, and deterministic stability guarantees, and compute projected tubes exactly by max-plus forward-backward recursions in O(TK^2) time for dense transitions. Posterior tube mass and HPD calibration are separate pathwise calculations approximated by FFBS. In a public bat-tracking application, robust foraging tube segments are enriched for feeding buzzes, whereas robust commuting segments are depleted: at eta = 0.005, enrichment is 2.25 with 95% bootstrap interval (1.73, 2.85) for robust foraging and 0.27 with interval (0.16, 0.44) for robust commuting.

In many engineering applications, a single high-fidelity model produces multiple quantities of interest (QoIs) under the same input parameters, e.g. finite element models of complex physical systems. To alleviate the high computational cost of direct model evaluations, surrogate models are widely used to construct efficient approximations of model responses. Naturally, the accuracy of surrogates strongly depends on the quality of the experimental design (ED). However, a single ED may not provide an adequate representation for all outputs simultaneously, especially when different outputs exhibit varying sensitivities to the input variables. A straightforward solution is to perform separate sampling for each output, but this results in increased sampling complexity and computational cost. From a statistical perspective, such an approach also ignores potential correlations among all outputs and may compromise data consistency. To address this issue, an adaptive sequential sampling method for constructing polynomial chaos expansion surrogate models is generalized for vector valued QoIs. The method sequentially selects new samples from a candidate pool based on their local contribution to the output variance, while balancing distance-based exploration of the input space and exploitation of aggregated variance information across all outputs. Its performance is compared with non-sequential Latin Hypercube Sampling through several numerical examples from engineering problems. Numerical results demonstrate that the proposed strategy improves both surrogate accuracy and stability, and provides a more reliable estimation of second-order statistics.

The nested sampling (NS) technique has gained widespread attention, particularly in cosmology and astronomy, due to its ability to efficiently explore high-likelihood regions - a feature akin to an implicit likelihood optimization that underlies its success. While the full theoretical derivation of NS is complex and involves several approximations, the central challenge lies in sampling from the likelihood-constrained priors, which is crucial for its performance. This work provides a comprehensive and detailed exposition of NS derivation, clarifying both its theoretical foundations and practical challenges. We provide a thorough description of the NS procedure, emphasizing both its strengths and potential limitations. In doing so, this work seeks to deepen understanding of the method and to foster the development of future enhancements, novel variants, and more efficient implementations across a wide range of scientific applications. Thus, the main contribution of this work is twofold: it serves both as a tutorial for newcomers to the field and as a critical review for experienced practitioners.

Markovian Whittle-Matérn fields have been convergently approximated by discrete Gauss Markov Random Fields (GMRFs) with sparse precision matrices using a Finite Element approximation of the two-parameter family, \[ (\kappa^2 - \Delta)^{\alpha/2} u = \mathcal{W}, \;\; \kappa \in \mathbb{R}, \; \alpha \in \mathbb{N}. \] of SPDEs. Using recent developements in the analysis of Discrete Exterior Calculus (DEC), we present a different, yet closely related, convergent GMRF approximation to these Matérn fields over complete, boundaryless Riemannian manifolds discretized as well-centered simplicial complexes. This convergent method (i) is agnostic to $\alpha, \kappa$ and thus allows a universal approximation scheme for the precision and covariance matrices of the entire $(\alpha, \kappa)$-family of GMRFs, so they may be inferred rather than guessed. (ii) inherently models pointwise and piecewise-smoothed measurements of a random field and approximates both equally well (iii) is computationally independent of the interpolants used - it suffers no overhead if one convergent interpolant were replaced with another suitable interpolant over the same mesh. Furthermore, we show that, on discretizations that are well-connected in a precise sense, and volume-concentrated, the precision matrices are spectral functions of a graph-laplacian. We provide a low rank approximator to the family of such Matérn GMRFs and mention a use case: reducing the number of measurements needed to model the GMRF by compressed-sensing.

We propose the data augmented bootstrap (DAB), a framework for constructing confidence intervals from approximately invariant transformations of the data. As special cases, DAB recovers popular methods that rely on exact group symmetries, such as conformal prediction, wild bootstrap for Maximum Mean Discrepancy U-statistics and the recently proposed SymmPI. Meanwhile, DAB also recovers the classical bootstrap method, which exploits the dataset's approximate invariance under uniform sampling of data indices as the dataset size grows. For all DAB methods, we establish theoretical coverage results that interpolate between finite-sample and asymptotic guarantees according to the strength of the invariance, and without assuming a group structure. The approximate invariance is measured in the Kolmogorov distance and, for statistics that satisfy Gaussian universality, reduces to conditional mean and variance matching. This allows us to incorporate data augmentation (DA), a widely used machine learning heuristic based on approximate invariances, into known statistical methods. We empirically test the performance of incorporating DA into bootstrap, wild bootstrap and conformal prediction for simulated settings as well as for image, language and scientific data.

We study parameter estimation for univariate stochastic differential equations with locally Lipschitz drift and Hölder continuous multiplicative diffusion, a class commonly arising in several applications. Existing inference methods typically rely on either the Euler-Maruyama discretisation, despite its lack of strong convergence and failure to preserve the state space, or on approximations, e.g. Gaussian approximation or truncation of Hermite's expansions, impacting on their stability and computational efficiency. We introduce the first explicit pseudo-likelihood estimators based on numerical splitting schemes that are both strong mean-square convergent and state space preserving for this class of SDEs. Our approach is based on a novel decomposition of the SDE that exploits reducibility and the Lamperti transform, leading to Lie-Trotter (LT) and Strang splitting schemes yielding explicit pseudo-likelihoods and maximum likelihood estimators based on them. We prove strong mean-square convergence, state space preservation, and improved robustness with respect to the discretisation step compared to Euler-Maruyama-based methods. We further establish consistency and asymptotic normality of the LT estimator. Because the proposed numerical scheme couples drift and diffusion parameters in the pseudo-likelihood, the asymptotic analysis requires new proof techniques. Extensive simulations demonstrate that the proposed estimators outperform existing methods in both accuracy and computational efficiency.

The mean-field Schrödinger bridge (MFSB) problem concerns designing a minimum-effort controller that guides a diffusion process with nonlocal interaction to reach a given distribution from another by a fixed deadline. Unlike the standard Schrödinger bridge, the dynamical constraint for MFSB is the mean-field limit of a population of interacting agents with controls. It serves as a natural model for large-scale multi-agent systems. The MFSB is computationally challenging because the nonlocal interaction makes the problem nonconvex. We propose a generalization of the Hopf-Cole transform for MFSB and, building on it, design a Sinkhorn-type recursive algorithm to solve the associated system of integro-PDEs. Under mild assumptions on the interaction potential, we discuss convergence guarantees for the proposed algorithm. We present numerical examples with repulsive and attractive interactions to illustrate the theoretical contributions.

We develop Riemannian approaches to variational autoencoders (VAEs) for PDE-type ambient data with regularizing geometric latent dynamics, which we refer to as VAE-DLM, or VAEs with dynamical latent manifolds. We redevelop the VAE framework such that manifold geometries, subject to our geometric flow, embedded in Euclidean space are learned in the intermediary latent space developed by encoders and decoders. By tailoring the geometric flow in which the latent space evolves, we induce latent geometric properties of our choosing, which are reflected in empirical performance. We reformulate the traditional evidence lower bound (ELBO) loss with a considerate choice of prior. We develop a linear geometric flow with a steady-state regularizing term. This flow requires only automatic differentiation of one time derivative, and can be solved in moderately high dimensions in a physics-informed approach, allowing more expressive latent representations. We discuss how this flow can be formulated as a gradient flow, and maintains entropy away from metric singularity. This, along with an eigenvalue penalization condition, helps ensure the manifold is sufficiently large in measure, nondegenerate, and a canonical geometry, which contribute to a robust representation. Our methods focus on the modified multi-layer perceptron architecture with tanh activations for the manifold encoder-decoder. We demonstrate, on our datasets of interest, our methods perform at least as well as the traditional VAE, and oftentimes better. Our methods can outperform this and a VAE endowed with our proposed architecture, frequently reducing out-of-distribution (OOD) error between 15% to 35% on select datasets. We highlight our method on ambient PDEs whose solutions maintain minimal variation in late times. We provide empirical justification towards how we can improve robust learning for external dynamics with VAEs.

In the study of complex systems, evaluating physical observables often requires sampling representative configurations via Monte Carlo techniques. These methods rely on repeated evaluations of the system's energy and force fields, which can become computationally expensive. To accelerate these simulations, deep learning models are increasingly employed as surrogate functions to approximate the energy landscape or force fields. However, such models introduce epistemic uncertainty in their predictions, which may propagate through the sampling process and affect the simulation's macroscopic behavior. In our work, we present the Penalty Ensemble Method (PEM) to quantify epistemic uncertainty and mitigate its impact on Monte Carlo sampling. Our approach introduces an uncertainty-aware modification of the Metropolis acceptance rule, which increases the rejection probability in regions of high uncertainty, thereby enhancing the reliability of the simulation outcomes.

In this paper, we study the problem of sampling from log-concave distributions supported on convex and compact sets, with a particular focus on the randomized midpoint discretization of both overdamped and kinetic Langevin diffusions in constrained domains. We revisit the proximal framework for handling constraints through projection operators and develop a more general formulation that encompasses Euclidean, Bregman, and Gauge projections. The resulting smooth approximation allows a unified and tractable analysis of Langevin algorithms and their variants under constraints. Within this framework, we establish convergence guarantees in Wasserstein-$q$ $(q\geqslant 1)$ distances between the smooth surrogate and the target distribution. We further derive complementary lower bounds, showing that the results are near-optimal in order. Building upon this tight approximation analysis, we obtain new convergence guarantees for the randomized midpoint Langevin algorithms and refined bounds for both vanilla and kinetic Langevin Monte Carlo methods under constraints, thereby advancing the theoretical understanding of constrained diffusion-based sampling.

We introduce Martingale Doppelgänger-Eval, a public shadow-market benchmark for auditing whether vision-language models (VLMs) use candlestick evidence rather than extrapolate past trends. The central difficulty is identification: on real market histories, chart evidence and trend are strongly coupled, so an observational score cannot determine whether a fluent technical-analysis narrative is grounded in local visual evidence. We prove this limitation formally: no evaluation functional computed from observational chart--label data can distinguish a grounded responder from a trend-shortcut responder under strong coupling, whereas matched evidence interventions separate the same responders at an exponential rate and trend--label swaps provide an independent shortcut stress test. The benchmark therefore evaluates frozen VLMs on rendered OHLCV charts under four controlled mechanisms: a martingale-null market, injected-alpha counterfactual pairs, trend-confounder swaps, and regime shifts. A structural behavioral model identifies null-market bias, trend sensitivity, evidence sensitivity, prompt/renderer fragility, and evidence faithfulness; the accompanying statistical toolkit provides minimum detectable effects, block-aware sequential testing for metered APIs, and an overlap-weighted artifact check. Across frozen commercial and open VLMs, the identified regression assigns large positive coefficients to past trend but evidence coefficients that are zero or opposite to the rule-implied sign. Matched-pair analyses show that models either ignore injected candlestick semantics or move opposite to the rule-implied direction conditional on responding. The benchmark isolates a failure mode that standard observational chart benchmarks cannot detect and gives a reusable audit template for time-series imagery with controllable label mechanisms.

Causal discovery seeks to uncover the causal dependencies among variables. For this purpose, we propose an algorithm called Tensor-based Second-order Causal Discovery (TSCD). Its input is a tensor obtained from the covariance matrices of observational and interventional data. Assuming the causal dependencies follow a linear structural equation model on a directed acyclic graph (DAG), TSCD outputs the DAG and the functions on its edges, requiring only that the noise variables are uncorrelated. We also implement a version of the approach for nonlinear models. Our focus on second-order statistics (via the covariance matrices) is motivated by their statistical and computational efficiency relative to higher-order moments, their identifiability relative to first-order statistics, and that they work regardless of whether the variables are Gaussian. We show that TSCD has identifiable causal order and parameters from a number of interventions that is logarithmic in the number of variables. Experiments show that TSCD is robust to noise, competitive with existing methods, and scales to hundreds of variables.

Constraint-based causal discovery relies on repeated conditional independence tests, but fast nonparametric tests often sacrifice calibration, especially when variables depend on the conditioning set through nonlinear relationships. We introduce BLITZ (Broad-to-Local Independence Testing via residualiZation), a nonparametric conditional independence test designed to run well under a second while maintaining the accuracy needed for the thousands of queries performed by constraint-based causal discovery algorithms. BLITZ first removes broad smooth dependence on the conditioning set using low-order polynomial regression, then applies a small nonlinear feature map and residualizes those features with shallow tree regressions. The resulting statistic tests residual cross-covariance, with a moment-matched chi-square approximation to the null distribution. We show theoretically that the two-stage design reduces the effective complexity faced by the tree residualizers, allowing shallow trees to control residual conditional-mean bias while avoiding excessive overfitting. In simulations, BLITZ provides better null calibration than fast kernel, random-feature, and regression-based competitors while remaining among the fastest methods tested. In causal discovery experiments on synthetic graphs and flow-cytometry data, BLITZ yields more reliable endpoint orientations among retained adjacencies and competitive structural recovery. These results suggest that broad-to-local residualization is a practical route to calibrated, scalable nonparametric conditional independence testing for causal discovery.

Agentic artificial intelligence systems introduce a new class of model risk. Unlike traditional predictive models, autonomous agents continuously acquire information, form beliefs regarding latent states of the environment, generate forecasts, select actions, and adapt their behavior over time. Existing validation methodologies focus primarily on predictive accuracy and therefore provide limited insight into the quality of the underlying decision process. This paper proposes a model validation framework for agentic AI based on Partially Observable Markov Decision Processes (POMDPs). The framework decomposes autonomous decision making into information, beliefs, forecasts, actions, and utility, allowing each component to be validated independently. Large language models (LLMs) are formalized as approximate Bayesian filtering operators, and a model-risk taxonomy is developed encompassing state-space, filtering, forecast, policy, utility-specification, and parameter risks. The model risk validation methodology is demonstrated through a portfolio-management case study in which an agent infers latent market regimes from market and macroeconomic information, generates belief-conditioned forecasts, and constructs portfolios using a Black--Litterman framework. Empirical validation combines performance analysis, belief calibration diagnostics, coverage tests, ablation studies, and parameter-sensitivity analysis. The results indicate that latent-state inference contributes independently to decision quality and that the principal conclusions remain robust across a broad range of parameter values. The principal contribution of the paper is a practical framework for extending established model risk management concepts to autonomous AI systems and providing a rigorous foundation for their validation, governance, and monitoring.

This paper is concerned with learning principal variations of random probability measures on $\mathbb{R}^m$ under the Wasserstein geometry. We introduce a new dynamical formulation to interpret the log-PCA, a linearized principal geodesic analysis, as a variational approach. Our differentiable version, termed as the Wasserstein Tangential PCA (WT-PCA), captures the local principal modes of geodesic variations of a (weighted) probability measure on the Wasserstein space via its covariance operator at barycenter. Based on the dynamical perspective and leveraging parallel transport structure of the optimal transport problems, we derive a general statistical convergence rate of the empirical WT-PCA when estimated from data in terms of the 2-Wasserstein distance between the population and empirical barycenter reference measures.

A certificate that removes outliers sees the data only through its low-degree moments, and an adversary exploits exactly this, hiding corruption where the clean data already looks typical, in the blind spot no bounded-degree test resolves. That blind spot turns out to have an exact size: the Christoffel function of the clean marginal, the very quantity modern data analysis thresholds to detect outliers, here read from the adversary's side as the corruption a bounded-degree certificate cannot remove. We turn this inversion into the organizing principle of the reweighted-hinge approach to robustly learning $\gamma$-margin halfspaces under malicious noise (Shen, 2025; Zeng and Shen, 2025): the governing resource is the Sum-of-Squares degree of the outlier-removal certificate, and the resolution principle states that the maximal corruption mass which can hide at a center $c$ from a degree-$2t$ certificate is exactly the Christoffel function $\lambda_{t+1}(c)$ of the clean marginal. Three consequences follow, all against the certificate method (not information-theoretic). A margin-degree tradeoff: certifying the dense pancake to error $\epsilon$ costs SoS degree $\Omega(\log(1/\epsilon))$ or margin $\Omega(\sqrt{\log(1/\epsilon)}/\sqrt{d})$, explaining why the $\log(1/\epsilon)$ margin Shen (2025) records is forced, with a weighted-Chebyshev reduction making the threshold $2t=\Theta((|c|/s)^2)$ tight modulo one classical weighted-extremal estimate. A degree-$2$ outlier barrier: the resolution principle realized as an explicit instance on which degree $2$ is stuck at $\eta^{1/2}$ while degree $4$ escapes, locating the method's small breakdown rate in the degree, not the analysis. And a degree-$2t$ algorithm tracing the frontier $\eta^{1-1/2t}$ (recovering Shen (2025) at $t=1$), whose gain is an explicit constant, capped by the pancake density and shown unimprovable by the degree-$2$ barrier.

Temporal difference (TD) learning with linear function approximation is a core method for policy evaluation. Its classical continuous-time description is an ordinary differential equation (ODE), which captures the asymptotic mean dynamics but neglects stochastic fluctuations determining the error floor. We introduce a stochastic differential equation (SDE) approximation for linear TD(0) under Markovian noise. The resulting model distinguishes the contraction dynamics governed by the projected Bellman operator from the influence of Markovian sampling. As a consequence, the model explains the constant-stepsize error floor through the interaction between Markovian long-run covariance and the contraction geometry of the projected Bellman operator.

We consider the concentration properties of functions of infinitely exchangeable random variables. By conditioning on the de Finetti directing measure, we show that the deviation of any function with bounded-difference constants $c_1, \dots, c_n$ decomposes into a conditional sampling fluctuation and a latent mixture fluctuation. When this latent mixture is $\sigma_{\mathrm{mix}}^2$-subgaussian, we establish a concentration inequality with an effective variance proxy of $\frac{1}{4}\sum_i c_i^2 + \sigma_{\mathrm{mix}}^2$. Crucially, we demonstrate that for zero-sum linear contrasts, such as the difference between a subsample mean and a full population mean, the latent mixture term cancels exactly. This cancellation yields a tight, mixture-free Hoeffding-type bound that provides a direct de Finetti mechanism for the infinite-extendibility limit of recent finite-exchangeable concentration results. We apply this framework to quantify uncertainty in composite AI benchmarks, such as MMLU, where question items naturally exhibit exchangeable dependence across domains. Our results provide both a domain-stratified hierarchical model for bounding the uncertainty of accuracy scores, and a distribution-free, cost-saving statistical guarantee for accurately estimating full benchmark scores from random subsets.

Motivated by the optimization of bounded binary black-box functions, we study the problem of learning polynomial surrogates over the Boolean hypercube. To ensure that optimizing the surrogate yields good solutions for the underlying objective, we require uniform $L_\infty$-error guarantees rather than the usual $L_2$-type guarantees. We characterize the minimax sample complexity of uniform estimation under subgaussian noise for two classes of bounded polynomials. First, for polynomials of degree at most $d$ on $n$ variables, the sample complexity scales as $n^{d+1}$. Second, for $s$-sparse Fourier-Walsh polynomials with $s \leq n$, it scales as $ns^2$. These rates differ structurally from the noiseless setting, where uniform exact recovery scales as $n^d$ and $ns$, respectively. Our lower bounds hold even for arbitrary adaptive learners, showing that the additional factors are intrinsic to the noisy cases. Standard Fourier-analysis tools for the $L_2$-norm do not naturally extend to the $L_\infty$-setting in a way that yields uniform guarantees. Our proofs overcome this difficulty by relying on suitably chosen auxiliary norms that serve as proxies for controlling the $L_\infty$-error. Together, our results provide a tight characterization of the sample complexity of learning optimization-safe polynomial surrogates.

Graph neural network architectures based on the graph Laplacian approximate the Laplace-Beltrami operator, thus limiting their application to isotropic operators. As a nonlinear alternative to the Laplace-Beltrami operator, we consider estimates of the Finsler Laplacian on point clouds sampled from a manifold. We prove that these discrete estimates converge to the true operator on the manifold as the number of point samples grows. Moreover, we show that this operator can be expressed as a graph neural network layer, which we use to define a family of Finslerian graph neural networks constrained to express Finsler geometry. We show that Finslerian graph neural networks recover the geometry underlying nonlinear diffusion equations in practice.

This paper builds a hierarchy of explicit, non-asymptotic tail bounds for the supremum of the Kostlan--Shub--Smale (KSS) random field on the sphere, and applies it to two problems: Spiked Tensor PCA and the landscape of the spherical $k$-spin model. For Tensor PCA, we study the non-asymptotic statistical limits of estimating a rank-$R$ symmetric signal tensor of order~$k\ge 3$ and dimension~$d\ge 3$ from a single Gaussian observation at signal-to-noise ratio~$\lambda$, through the \emph{profile maximum likelihood estimator}, the MLE restricted to normalized rank-$R$ tensors of coherence at least~$\kappa$. Our analysis uses a single reduction: a deterministic geometric inequality (the Tube Method) and a rank-reduction step bound the estimation error by the supremum of the canonical KSS field, which the Kac--Rice formula turns into a Gaussian integral against the expected absolute characteristic polynomial of a shifted Gaussian Orthogonal Ensemble, controlled in turn by the four explicit tail bounds of our hierarchy (three from a Mehta--Fyodorov representation, one from a Ben Arous--Dembo--Guionnet large deviation). The same reduction yields two results, each with explicit constants. For estimation, a finite-$(k,d)$ error bound recovers the asymptotically optimal rate~$\sqrt{d\log k}$ of Perry, Wein and Bandeira, with explicit dependence on the rank~$R$ and the coherence~$\kappa$. For the landscape, a two-sided non-asymptotic bracketing of the annealed complexity of the spherical $k$-spin Hamiltonian recovers the Auffinger--Ben Arous--Černý complexity function in the high-dimensional limit.

Adaptive optimizers combining preconditioning, momentum, and weight decay (Adam and AdamW) are, under Polyak-Ruppert averaging, candidate engines for one-pass inference. Does the averaged iterate keep the classical Polyak-Ruppert central limit theorem (CLT), with sandwich covariance $H^{-1}SH^{-1}$ (Hessian $H$, gradient covariance $S$), under momentum and non-convergent preconditioning? The preconditioner-only analysis does not carry over: with momentum the canonical decomposition collapses to a tautology. Treating the augmented state (iterate, momentum buffer) as a time-varying linear stochastic approximation (SA), we prove (under local stabilization) positive drift stability, a non-autonomous Polyak-Ruppert CLT, and a projection identity. The upshot: the iterate-marginal covariance is exactly the plain stochastic gradient descent (SGD) sandwich $H^{-1}SH^{-1}$, so the adaptivity is asymptotically invisible. This holds for SA-Adam (sub-linearly vanishing momentum gain, $\gamma\in(\alpha,1)$; the sub-linear regime is essential), not constant-$\beta$ deployed Adam. Coupled $L_2$ weight decay yields the ridge-penalized sandwich, extending one-pass inference to regularized problems.

We develop Hamiltonian dynamics-based algorithms for smooth convex optimization that achieve accelerated rates of convergence. By exploiting contraction of averaged Hamiltonian flow trajectories rather than requiring contraction at trajectory endpoints, we show that Hamiltonian dynamics-based optimization methods admit deterministic and accelerated convergence guarantees, extending prior work that is limited to quadratic objectives or holds only in expectation. We analyze an idealized continuous-time algorithm and derive practical discrete-time implementations with optimal first-order complexity, thereby establishing Hamiltonian dynamics as a useful algorithmic primitive for deterministic accelerated convex optimization.

Evaluating representation similarity is fundamental to representation learning. However, existing metrics suffer from significant limitations: they lack interpretability due to shifting baselines, lack robustness to outliers, and are computationally intractable for large datasets, forcing reliance on heuristic approximations. To address this, we develop an ordinal-similarity framework, instantiated by the Triplet (TSI) and Quadruplet (QSI) Similarity Indices, which measure alignment by quantifying the consistency of ordinal relationships. We theoretically demonstrate this formulation is inherently interpretable, robust to outliers, and computationally efficient. Finally, we establish a formal equivalence between TSI and local neighborhood alignment, measured by Mutual Nearest Neighbors. Empirically, we validate these properties and show that ordinal similarity offers a scalable approach to measuring alignment, enabling practitioners to better understand and design representations.

We develop a general framework for analyzing representation costs of parametric data-fitting methods through their parameter-space regularizers. From this abstract perspective, we define representation costs for arbitrary parametric models and reveal their induced (native) function spaces. This unifies recent function-space views of data-fitting methods. We also prove that many natural results hold in this abstract setting, including representer theorems for parametric methods on their native spaces. The framework also rigorously connects parametric methods with their equivalent nonparametric descriptions under sufficient overparameterization. Classical methods and their native spaces, such as kernel methods / reproducing kernel Hilbert spaces, wavelets / Besov spaces, and shallow neural networks / variation spaces emerge as special cases of our abstract framework. A byproduct of "axiomatizing" the study of representation costs is that we also immediately obtain new results for deep neural networks: For depth-$L$ feedforward ReLU networks, their induced native spaces are $p$-normable quasi-Banach spaces with $p = 2/L$. This reveals that the inductive bias of deep neural networks (as given by the representation cost) cannot be captured by norms for depths $L > 2$.

Recent work in random matrix theory (RMT) has developed the notion of deterministic equivalents: typically linear surrogate models that approximate the spectral behavior of large nonlinear random matrices, such as nonlinear feature maps in neural networks (NNs). Such equivalents make theoretical predictions tractable by reducing a complex model to a simpler one with properties that fall under the umbrella of classical RMT tools. However, this leaves open the question of whether this idealized linear equivalence remains meaningful for classification of high-dimensional nonlinearly separable data. Motivated by this, we consider the conjugate kernel (CK), which is the nonlinear feature map of a one-layer feedforward NN, under a canonical nonlinearly separable dataset for the XOR problem; and we use the study of informative outlier eigenvalues in the CK and whether their corresponding eigenvectors asymptotically align with XOR labels as a proxy for nonlinear learnability. We develop a robust quadratic equivalent of the CK matrix that enables a precise analysis of emergent informative spikes, as one modifies various knobs common in ML practice: sample complexity, signal-to-noise ratio (SNR), nonlinear activation choice, and pretrained features. We identify regimes in which these knobs move the CK beyond the linear equivalent and produce BBP-type transitions to label-aligned outlier eigenspaces. Our analysis helps bring deterministic-equivalence tools from RMT to bear on problems of practical relevance in ML.

Conformal prediction is a framework for providing prediction intervals with distribution-free validity, guaranteeing predictive coverage for data drawn from any distribution. Its two main variants are full conformal prediction and split conformal prediction (also called transductive and inductive). Full conformal prediction is widely considered to be statistically more efficient (since split conformal prediction requires data splitting, and therefore can lead to wider prediction intervals due to the resulting loss in sample size), but its implementation is computationally prohibitive, as it requires the underlying model to be refit for every candidate value in the response space. Existing computational shortcuts, such as using a discrete grid of values to approximate the full conformal prediction construction, frequently lack theoretical guarantees on marginal coverage and can fail in practice. To address this limitation, we introduce a novel class of approximations to the full conformal prediction method, based on the idea of \emph{tournaments}, which enables the construction of prediction sets with a rigorous marginal coverage guarantee of $1-2\alpha$. Under stability conditions, the theoretical coverage guarantee tightens to approximately $1-\alpha$. This new framework generalizes the existing method of leave-one-out cross-conformal prediction, while allowing for flexible use of various existing approximation strategies.

Local prediction-error-based curiosity rewards focus on the current transition without considering the world model's cumulative prediction error across all visited transitions. We introduce Curiosity-Critic, which grounds its intrinsic reward in the improvement of this cumulative objective, and show that it admits a tractable per-step surrogate: the difference between the current prediction error and the asymptotic error baseline of the current state transition. We estimate this error baseline online with a learned critic co-trained alongside the world model; since the critic only has to learn how hard a transition is to predict, its estimate of the irreducible noise floor converges well before the world model saturates, redirecting exploration toward learnable transitions. The reward is higher for learnable transitions and collapses toward zero for stochastic ones, thereby separating epistemic (reducible) from aleatoric (irreducible) prediction error online. Prior prediction-error curiosity formulations, from Schmidhuber (1991) to learned-feature-space variants, emerge as special cases corresponding to specific approximations of this error baseline. Experiments on a stochastic grid world show that Curiosity-Critic outperforms prediction-error, visitation-count, and Random Network Distillation methods in training speed and final world model accuracy.

Global variable importance measures are commonly used to interpret the results of machine learning models. Local variable importance techniques assess how variables contribute to individual observations. Current, popular methods, including LIME and SHAP, provide useful measures of feature contribution in the prediction space, while leaving opportunities for improved characterization of local structure in the model loss space. Additionally, they are not natively adapted for multi-class classification problems. We propose a new model-agnostic method for calculating local variable importance, CLIQUE, that highlights locally dependent relationships, provides improved stability over permutation-based methods, and can be directly applied to multi-class classification problems. Simulated and real-world examples show that CLIQUE emphasizes locally dependent information, captures interaction behavior beyond what can be evaluated by correlations, and assigns zero importance in regions where the response is invariant to changes in variables.

Maximum inner product search (MIPS) is a crucial subroutine in machine learning, requiring the identification of a vector taken within a database (the keys) that best aligns with a given query. We propose amortized MIPS: a regression-based approach that trains neural networks to directly predict MIPS solutions, amortizing the cost of repeatedly solving MIPS for queries drawn from a known distribution over a fixed key database. Our key insight is that the MIPS value function is the \emph{support} function of the set of keys, a well-studied convex function whose gradient yields the optimal key. This motivates two complementary amortized models: SupportNet, an input-convex neural network trained to regress the support function, and KeyNet, a vector-valued network that directly regresses the optimal key. SupportNet can serve as a cluster router, steering queries toward relevant database partitions, while KeyNet can be used as a drop-in replacement for the original query, fed directly to off-the-shelf indexing pipelines. Our experiments on the BEIR benchmark show that, for document embeddings, learned \SupportNet{}s and \KeyNet{}s significantly improve IVF match rates when accounting for compute effort, whether measured in FLOPs, number of probes, or wall-clock time. Our code is available at: this https URL.

We consider the problem of regularized best-response max-regret minimization in online RLHF under general preferences and bandit feedback. While various regularizers are utilized to robustify alignment, known polylogarithmic regret guarantees remain heavily specific to KL. To investigate whether such fast rates extend beyond KL, we adopt the Generalized Bilinear Preference Model (GBPM) -- capturing intransitive preferences over $d$-dimensional item-wise features via a rank-$2r$ skew-symmetric matrix -- to isolate the impact of generic regularization. Crucially, under GBPM, we prove that the dual gap of any greedy policy is bounded by the squared estimation error, derived using \emph{only} strong convexity and skew-symmetry. Under a feature coverage assumption, we establish a \emph{generic} polylogarithmic regret of $\tilde{\mathcal{O}}(\eta d^4 C_{\min}^{-1} (\log T)^2 \wedge d^2 C_{\min}^{-1/2} \sqrt{T})$ with Greedy Sampling, and a dimension-wise improved regret (for well-conditioned arm-sets) of $\tilde{\mathcal{O}}(C_{\min}^{-2} \sqrt{\eta r T} \wedge r^{1/3} C_{\min}^{-4/3} T^{2/3})$ with Explore-Then-Commit, where $\eta^{-1}$ is the regularization coefficient, $T$ is the time horizon, and $C_{\min}$ is an arm-set dependent quantity. This demonstrates that ``fast'' regrets are not KL-specific, but rather a fundamental consequence of generic strongly convex geometry.

Sparse autoencoders (SAEs) are widely used to extract human-interpretable features from neural network activations, but their learned features can vary substantially across random seeds and training choices. To improve stability, we studied weight regularization by adding L1 or L2 penalties on encoder and decoder weights, and evaluate how regularization interacts with common SAE training defaults. On MNIST, we observe that L2 weight regularization produces a core of highly aligned features and, when combined with tied initialization and unit-norm decoder constraints, it dramatically increases cross-seed feature consistency. For TopK SAEs trained on language model activations (Pythia-70M-deduped), adding a small L2 weight penalty increased the fraction of features shared across three random seeds and roughly doubles steering success rates, while leaving the mean of automated interpretability scores essentially unchanged. Finally, in the regularized setting, activation steering success becomes better predicted by auto-interpretability scores, suggesting that regularization can align text-based feature explanations with functional controllability.

Data collection is a critical component of modern statistical and machine learning pipelines, particularly when data must be gathered from multiple heterogeneous sources to study a target population of interest. In many use cases, such as medical studies or political polling, different sources incur different sampling costs. Observations often have associated group identities - for example, health markers, demographics, or political affiliations - and the relative composition of these groups may differ substantially, both among the source populations and between sources and target population. In this work, we study multi-source data collection under a fixed budget, focusing on the estimation of population means and group-conditional means. We show that naive data collection strategies (e.g. attempting to "match" the target distribution) or relying on standard estimators (e.g. sample mean) can be highly suboptimal. Instead, we develop a sampling plan which maximizes the effective sample size - the total sample size divided by $D_{\chi^2}(q\mid\mid\overline{p}) + 1$, where $q$ is the target distribution, $\overline{p}$ is the aggregated source distribution, and $D_{\chi^2}$ is the $\chi^2$-divergence. We pair this sampling plan with a classical post-stratification estimator and upper bound its risk. We provide matching lower bounds, establishing that our approach achieves the budgeted minimax optimal risk. Our techniques also extend to prediction problems when minimizing the excess risk, providing a principled approach to multi-source learning with costly and heterogeneous data sources.

Thompson sampling (TS) is widely used for stochastic multi-armed bandits, yet its inferential properties under adaptive data collection are subtle. Classical asymptotic theory for sample means can fail because arm-specific sample sizes are random and coupled with the rewards through the action-selection rule. We study adaptive inference for Thompson sampling with Gaussian randomized indices in $K$-armed stochastic bandits with independent sub-Gaussian reward noises, and identify \emph{optimism} as a key mechanism for restoring \emph{stability}, meaning that each arm's pull count concentrates around a deterministic scale. This stability yields asymptotically valid Wald inference despite adaptive sampling. First, we prove that variance-inflated TS is stable for any $K \ge 2$, including the challenging regime where multiple arms are optimal, with asymptotically uniform allocation over optimal arms and sharp logarithmic pull-count asymptotics for suboptimal arms. This resolves the $K$-armed extension question raised by \citet{halder2025stable}, using new winner-map and Lyapunov-drift techniques to control allocation among multiple optimal arms. Second, we analyze an alternative optimistic modification that keeps the Gaussian index variance unchanged but adds an explicit mean bonus to the index center, and establish a similar stability conclusion. In summary, suitably implemented optimism stabilizes Thompson sampling and enables asymptotically valid Wald inference in multi-armed bandits, while incurring only a mild additional regret cost.

Fast computation of a matrix product $W^\top X$ is a workhorse of modern LLMs. To make their deployment more efficient, a popular approach is that of using a low-precision approximation $\widehat W$ in place of true $W$ (``weight-only quantization''). Information theory demonstrates that an optimal algorithm for reducing precision of $W$ depends on the (second order) statistics of $X$ and requires a careful alignment of vector quantization codebook with PCA directions of $X$ (a process known as ``waterfilling allocation''). Dependence of the codebook on statistics of $X$, however, is highly impractical. This paper proves that there exist a universal codebook that is simultaneously near-optimal for all possible statistics of $X$, in the sense of being at least as good as an $X$-adapted waterfilling codebook with rate reduced by 0.11 bit per dimension in the case when $W$ is Gaussian. Such universal codebook would be an ideal candidate for the low-precision storage format, a topic of active modern research, but alas the existence proof is non-constructive. Equivalently, our result shows existence of a net in $\mathbb{R}^n$ that is a nearly-optimal covering of a sphere simultaneously with respect to all Hilbert norms.

Conformal prediction(CP) has become a cornerstone of distribution-free uncertainty quantification, conventionally evaluated by its coverage and interval length. This work critically examines the sufficiency of these standard metrics. We demonstrate that the interval length might be deceptively improved through a counter-intuitive approach termed Prejudicial Trick(PT), while the coverage remains valid. Specifically, for any given test sample, PT probabilistically returns an interval, which is either null or constructed using an adjusted confidence level, thereby preserving marginal coverage. While PT potentially yields a deceptively lower interval length, it introduces practical vulnerabilities: the same input can yield completely different prediction intervals across repeated runs of the algorithm. We formally derive the conditions under which PT achieves these misleading improvements and provide extensive empirical evidence across various regression and classification tasks. Furthermore, we introduce a new metric interval stability which helps detect whether a new CP method implicitly improves the length based on such PT-like techniques. Code is available at this https URL.

The data processing inequality is an information-theoretic principle stating that the information content of a signal cannot be increased by processing the observations. In particular, it suggests that there is no benefit in enhancing the signal or encoding it before addressing a classification problem. This assertion can be proven to be true for the case of the optimal Bayes classifier. However, in practice, it is common to perform "low-level" tasks before "high-level" downstream tasks despite the overwhelming capabilities of modern deep neural networks. In this paper, we aim to understand when and why low-level processing can be beneficial for classification. We present a comprehensive theoretical study of a binary classification setup, where we consider a classifier that is tightly connected to the optimal Bayes classifier and converges to it as the number of training samples increases. We prove that for any finite number of training samples, there exists a pre-classification processing that improves the classification accuracy. We also explore the effect of class separation, training set size, and class balance on the relative gain from this procedure. We support our theory with an empirical investigation of the theoretical setup. Finally, we conduct an empirical study where we investigate the effect of denoising and encoding on the performance of practical deep classifiers on benchmark datasets. Specifically, we vary the size and class distribution of the training set, and the noise level, and demonstrate trends that are consistent with our theoretical results.

In this work, we investigate the existence and effect of percolation in training deep Neural Networks (NNs) with dropout. Dropout methods are regularisation techniques for training NNs, first introduced by G. Hinton et al. (2012). These methods temporarily remove connections in the NN, randomly at each stage of training, and update the remaining subnetwork with Stochastic Gradient Descent (SGD). The process of removing connections from a network at random is similar to percolation, a paradigm model of statistical physics. If dropout were to remove enough connections such that there is no path between the input and output of the NN, then the NN could not make predictions informed by the data. We study new percolation models that mimic dropout in NNs and characterise the relationship between network topology and this path problem. The theory shows the existence of a percolative effect in dropout. We also show that this percolative effect can cause a breakdown when training NNs without biases with dropout; and we argue heuristically that this breakdown extends to NNs with biases.

Softmax attention is a central component of transformer architectures, yet its nonlinear structure poses significant challenges for theoretical analysis. We develop a unified, measure-based framework for studying single-layer softmax attention under both finite and infinite prompts. For i.i.d. Gaussian inputs, we lean on the fact that the softmax operator converges in the infinite-prompt limit to a linear operator acting on the underlying input-token measure. Building on this insight, we establish non-asymptotic concentration bounds for the output and gradient of softmax attention, quantifying how rapidly the finite-prompt model approaches its infinite-prompt counterpart, and prove that this concentration remains stable along the entire training trajectory in general in-context learning settings with sub-Gaussian tokens. In the case of in-context linear regression, we use the tractable infinite-prompt dynamics to analyze training at finite prompt length. Our results allow optimization analyses developed for linear attention to transfer directly to softmax attention when prompts are sufficiently long, showing that large-prompt softmax attention inherits the analytical structure of its linear counterpart. This, in turn, provides a principled and broadly applicable toolkit for studying the training dynamics and statistical behavior of softmax attention layers in large prompt regimes.

This work studies the statistical implications of using features comprised of general linear combinations of covariates to partition the data in randomized decision tree and forest regression algorithms. Using random tessellation theory in stochastic geometry, we provide a theoretical analysis of a class of efficiently generated random tree and forest estimators that allow for oblique splits along such features. We call these estimators oblique Mondrian trees and forests, as the trees are generated by first selecting a set of features from linear combinations of the covariates and then running a Mondrian process that hierarchically partitions the data along these features. Quadratic risk bounds and convergence rates are obtained for the flexible function class of multi-index models for dimension reduction, where the output is assumed to depend on a low-dimensional relevant feature subspace of the input domain. The results highlight how the risk of these estimators depends on the choice of features and quantify how robust the risk is with respect to error between the selected features along which the data is split and the true relevant feature subspace. The asymptotic analysis also provides conditions on the convergence rate a set of estimated relevant features must satisfy for oblique Mondrian estimators to obtain minimax optimal rates of convergence with respect to the dimension of the relevant feature subspace. Additionally, a lower bound on the risk of axis-aligned Mondrian trees (where features are restricted to the set of covariates) is obtained, proving that these estimators are suboptimal for general ridge functions, no matter how the distribution over the covariates used to divide the data at each tree node is weighted.

We investigate the fundamental problem of leveraging offline data to accelerate online reinforcement learning - a direction with strong potential but limited theoretical grounding. Our study centers on how to \emph{learn} and \emph{apply} value envelopes within this context. To this end, we introduce a principled two-stage framework: the first stage uses offline data to derive upper and lower bounds on value functions, while the second incorporates these learned bounds into online algorithms. Our method extends prior work by decoupling the upper and lower bounds, enabling more flexible and tighter approximations. In contrast to approaches that rely on fixed shaping functions, our envelopes are data-driven and explicitly modeled as random variables, with a filtration argument ensuring independence across phases. The analysis establishes high-probability regret bounds determined by two interpretable quantities, thereby providing a formal bridge between offline pre-training and online fine-tuning. Empirical results on tabular MDPs demonstrate substantial regret reductions compared with both UCBVI and prior methods while remaining competitive with related approaches.

Given a trained neural network, can any specified output be generated by some input? Equivalently, does the network correspond to a function that is surjective? In generative models, surjectivity implies that any output, including harmful or undesirable content, can in principle be generated by the networks, raising concerns about model safety and jailbreak vulnerabilities. In this paper, we prove that many fundamental building blocks of modern neural architectures, such as networks with pre-layer normalization and linear-attention modules, are almost always surjective. As corollaries, widely used generative frameworks, including GPT-style transformers and diffusion models with deterministic ODE solvers, admit inverse mappings for arbitrary outputs. By studying surjectivity of these modern and commonly used neural architectures, we contribute a formalism that sheds light on their unavoidable vulnerability to a broad class of adversarial attacks.

Recent studies identified an intriguing phenomenon in recursive generative model training known as model collapse, where models trained on data generated by previous models exhibit severe performance degradation. Addressing this issue and developing more effective training strategies have become central challenges in generative model research. In this paper, we investigate this phenomenon within a novel framework, where generative models are iteratively trained on a combination of newly collected real data and synthetic data from the previous training step. To develop an optimal training strategy for integrating real and synthetic data, we evaluate the performance of a weighted training scheme in various scenarios, including Gaussian distribution estimation, generalized linear models, and nonparametric estimation. We theoretically characterize the impact of the mixing proportion and weighting scheme of synthetic data on the final model's performance. Our key finding is that, across different settings, the optimal weighting scheme under different proportions of synthetic data asymptotically follows a unified expression, revealing a fundamental trade-off between leveraging synthetic data and model performance. In some cases, the optimal weight assigned to real data corresponds to the reciprocal of the golden ratio. Finally, we validate our theoretical results on extensive simulated datasets and a real tabular dataset.

We introduce an amortized Bayesian framework for spatial boundary detection that generalizes posterior inference across areal graphs with varying numbers of regions and diverse adjacency structures. The underlying model couples a Poisson count likelihood with a covariate-driven rule to interrupt smoothing across dissimilar neighboring areas, utilizing a directed acyclic graph autoregressive (DAGAR) prior to capture residual spatial dependence. To approximate the target posterior distribution, a neural engine is trained on simulated maps: a permutation-invariant summary network encodes graph-aware representations of the observed counts, offsets, covariates, and adjacency matrices, while a conditional normalizing flow generates the approximate posterior draws. Simulation studies demonstrate accurate parameter recovery, near-nominal interval coverage, well-calibrated posterior predictive behavior, and informative posterior boundary probabilities. Benchmarking against Markov chain Monte Carlo (MCMC) confirms close agreement regarding primary boundary evidence, and an ablation study validates the inclusion of model-guided graph summaries. Finally, applications to Glasgow respiratory disease and California lung cancer data demonstrate that a single trained neural engine can be seamlessly deployed across real-world maps with distinct graph structures, yielding boundary conclusions consistent with established localized smoothing analyses.

Alzheimer's disease is characterized by the progressive accumulation of amyloid-$\beta$ and tau followed years later by cognitive impairment. Despite this established motif, substantial subject-level variability exists in the age of pathological progression and the onset of cognitive symptoms. To understand the source of this variation, subjects must be aligned across heterogeneous disease timelines via frameworks that jointly model disease progression and time to cognitive impairment with reference to landmark positivity thresholds. Existing neurodegenerative disease progression models rely on restrictive parametric forms, fail to anchor disease timelines to positivity thresholds, and decouple biomarker trajectories from cognitive survival endpoints. To address these limitations, we introduce the Bayesian Threshold-Aligned Joint Disease Progression Model (B-TAJ DPM). This generative, semi-parametric framework models multivariate disease progression trajectories over latent disease timelines anchored at landmark positivity thresholds. Crucially, the framework integrates a survival model to link pathological progression to cognitive impairment. Posterior inference and posterior predictions for unseen subjects are carried out in open-source software. Simulation studies demonstrate excellent estimation accuracy and interval coverage. When applied to Alzheimer's Disease Neuroimaging Initiative data, B-TAJ DPM characterizes non-linear progression patterns, quantifies subject-level variation in positivity age, and reveals links between age of tau positivity and acceleration of cognitive impairment.

The influence of environmental exposures, such as air pollution, on human health has become increasingly recognized. A growing body of evidence suggests that the microbiome may mediate these effects, explaining the relationship between the environment and host biology. However, the impact of environmental exposures on the microbiome is not yet fully understood, and statistical modeling in this context is challenged by complex dependency structures. In particular, microbiome data exhibit spatial dependencies across sampling regions as well as ecological correlations among microbial taxa, which, if ignored, can substantially reduce detection power, leading to missed true signals. We introduce a novel spatial mixed modeling framework for microbiome data that accounts for both region-level spatial dependency and taxon-level ecological dependency using conditional autoregressive priors. Through simulations, we demonstrate that this framework outperforms existing methods that ignore such dependencies, by achieving high detection power in feature selection while maintaining low false positive rates and reduced mean squared error in estimation. Applied to two real studies-data from Food and Microbiome Longitudinal Investigation study and lung microbiome dataset-with fine particulate matter (PM_2.5) exposures, our model identified genera, which are known to be involved in pollution-related health outcomes, as well as novel taxa that may mediate host responses to air pollution. This novel approach offers a powerful and flexible tool for uncovering biologically meaningful associations in complex environmental data.

Subgroup analysis is routinely used in randomized controlled trials to examine whether treatment effects are homogeneous across patient subgroups or differ because of treatment-effect heterogeneity. In this paper, we investigate the properties of the odds ratio and the relative response in subgroup analyses with binary outcomes, extending previous work with new theoretical insights and methodological developments. We establish several new theorems that characterize how the odds ratio for the overall population changes in both magnitude and direction when two subgroups are combined. These results further confirm that the odds ratio is inappropriate as an efficacy measure in this subgroup setting, whereas the relative response is appropriate. We also present the formal relationship between the odds ratio and the relative response, and clarify their differences in terms of the logic-respecting property, that is, whether the overall efficacy lies between the subgroup efficacies, and the dilution property, that is, whether mixing subgroups moves the overall odds ratio toward 1. Although the odds ratio is generally not logic-respecting, it may behave approximately like a logic-respecting efficacy measure under certain conditions. To illustrate our findings, we present an illustrative example based on clinical trial data and discuss its implications for subgroup analysis in randomized controlled trials.

Binary endpoints are common in clinical trials and conditional odds ratios have traditionally been used to assess treatment effects. However, the interpretation of odds ratios is difficult, they are non-collapsible and rely on strong assumptions in order to be a relevant overall summary measure for the trial. As an alternative, risk differences have gained increasing prominence as a more interpretable, clinically meaningful and assumption-lean measure of treatment effects. This shift has also been motivated by new regulatory guidance, which emphasizes the relevance of marginal estimands and encourages covariate adjustment. Yet, covariate-adjusted inference for risk differences, particularly in smaller samples, has methodological subtleties and lacks well-established best practices. We conduct a simulation study comparing methods for estimating and testing risk differences in small-sample ($N \leq 150$) randomized clinical trials with prognostic categorical baseline covariates, focusing on exact unconditional tests, Mantel-Haenszel methods, and $g$-computation (standardization) approaches. We find that several $g$-computation approaches exhibit inflated Type-I error in very small samples when standard Wald-type inference is applied, whereas robust or penalized variants improve error control at the expense of power. Classical methods such as the Mantel-Haenszel and Suissa-Shuster tests remain robust but may forgo efficiency gains from covariate adjustment. Overall, our results indicate that much of the observed Type-I error inflation reflects misalignment between estimand and variance estimation rather than small sample size alone. Based on these results, we provide practical recommendations to guide method selection that align the estimand, variance estimation, and inferential target.

We propose a novel Phase I intra-patient dose-escalation design tailored for multi-cycle immunotherapy settings, in which toxicity at a fixed dose level is clinically expected to decrease over successive treatment cycles. This design was motivated by a phase I trial of CAR T cell therapy, an emerging cellular immunotherapy with established applications in cancer and growing investigation in autoimmune disease. The design is intended for settings in which nonincreasing cycle-specific toxicity assumption is clinically justified. Specifically, we build on the extrapolation property of the modified curve-free Bayesian decision-theoretic (c-CFBD) design for two-agent trials (Xu, et al. 2025), treating treatment cycle as a second dimension. By redefining the partial order, the c-CFBD framework can accommodate the reduction in toxicity across cycles. The proposed design adopts a two-stage structure: an initial accelerated titration stage to rapidly explore dose levels, followed by a c-CFBD stage to improve safety and estimate the cycle-specific maximum tolerated dose sequence. Simulation studies across a range of scenarios demonstrate favorable operating characteristics.

Most functional magnetic resonance imaging studies rely on estimates of hierarchically organized functional brain networks whose segregation and integration reflect the cognitive and behavioral changes in humans. However, most existing methods for estimating the community structure of networks from both individual and group-level analysis methods do not account for the variability between subjects. In this paper, we develop a new multilayer community detection method based on Bayesian latent block model (LBM). The method can robustly detect the community structure of weighted functional networks with an unknown number of communities at both individual and group levels and retain the variability of the individual networks. For validation, we propose a new community structure-based multivariate Gaussian generative model to simulate synthetic signal. Our simulation study shows that the community memberships estimated by hierarchical Bayesian inference are consistent with the predefined node labels in the generative model. The method is also tested via split-half reproducibility using working memory task fMRI data of 100 unrelated healthy subjects from the Human Connectome Project. Analyses using both synthetic and real data show that our proposed method is more accurate and reliable compared with the commonly used (multilayer) modularity models.

Explicit modelling of between-study heterogeneity is essential in network meta-analysis (NMA) to ensure valid inference and avoid overstating precision. While the additive random-effects (RE) model is the conventional approach, the multiplicative-effect (ME) model remains underexplored. The ME model inflates within-study variances by a common factor estimated via weighted least squares, yielding identical point estimates to a fixed-effect model while inflating confidence intervals. We empirically compared RE and ME models across NMAs of two-arm studies with significant heterogeneity from the nmadb database, assessing model fit using the Akaike Information Criterion. The ME model often provided comparable or better fit to the RE model. Case studies further revealed that RE models are sensitive to extreme and imprecise observations, whereas ME models assign less weight to such observations and hence exhibit greater robustness to publication bias. Our results suggest that the ME model warrant consideration alongside conventional RE model in NMA practice.

Survival analysis provides statistical methods to model the time until an event occurs. Reporting delays arise when event times are not observed at their occurrence but are only revealed upon reporting. This issue is particularly critical for timely risk evaluation when the observation window is short due to administrative censoring. In this study, we incorporate right-censored reporting delays by jointly modeling parametric hazards for the event and reporting processes. We then construct a consistent estimator for the model parameters and develop a Monte Carlo expectation-maximization algorithm to compute it. To address the challenges posed by administrative censoring, we leverage these findings and propose a transfer-learning procedure. Experimental results demonstrate that our method improves the accuracy of timely risk evaluation under administrative censoring.

Topological analyses of brain functional connectivity usually reduce each pair of channels to a single scalar dependence, typically the Pearson correlation, and so cannot resolve the frequency-specific synchronisation that organises electrophysiology. We propose a topological summary that keeps the frequency information. The spectral landscape indexes the persistence landscape of Bubenik (2015) by Fourier frequency, building each filtration from a coherence-based distance, so that it is a function of both the filtration scale and the frequency. It is Lipschitz-stable in the coherence matrix and feeds a functional two-sample test over a chosen frequency band, whose limiting null distribution and consistency follow from standard functional-data arguments. In simulations the test recovers a topological difference in the band where it lives while holding its nominal level under the null. Applied to electroencephalography from 53 control and 51 ADHD children, a global test rejects equality of the two groups' cycle topology at the 95% level (p = 0.019); a band-by-band follow-up localises the difference to the gamma and theta bands, although none survives family-wise correction at this sample size. The pattern is consistent with the established role of these bands in ADHD.

Understanding how extreme price movements propagate across financial and energy markets is critical for risk management and regulatory design in the EU Emissions Trading System (EU ETS). We apply Hüsler-Reiss graphical models of extremes to a system of 20 daily variables centred on EU allowances futures across Phases 3 and 4 of the EU ETS (2013--2025), with a Gaussian graphical model as the average-dependence baseline. The tail networks are structurally distinct from the average dependence network: substantially denser, organized around different central nodes, and governed by within-sector homophily that binds sector boundaries more tightly than at the average-dependence level. EU allowances futures are peripheral in the standard graphical model but achieve the highest centrality in the tail networks, while equity indices and major FX pairs follow the opposite trajectory. Exponential random graph models confirm equity and FX peripherality in tail networks across all sample periods and identify triadic closure during market downturns as a Phase~3 phenomenon that vanishes in Phase~4. The phase transition restructures the tail network without thinning it: average dependence contracts sharply while tail dependence persists, and crash contagion shifts from clustered to diffuse propagation. These findings have direct implications for hedge construction by compliance entities, stress-test calibration by regulators, and the design of systemic-risk monitoring tools for EU ETS markets.

This paper presents a comprehensive study on the use of ensemble Reinforcement Learning (RL) models in financial trading strategies, leveraging classifier models to enhance performance. By combining RL algorithms such as A2C, PPO, and SAC with traditional classifiers like Support Vector Machines (SVM), Decision Trees, and Logistic Regression, we investigate how different classifier groups can be integrated to improve risk-return trade-offs. The study evaluates the effectiveness of various ensemble methods, comparing them with individual RL models across key financial metrics, including Cumulative Returns, Sharpe Ratios (SR), Calmar Ratios, and Maximum Drawdown (MDD). Our original experimental results demonstrate that ensemble methods often outperform base models in terms of risk-adjusted returns, providing better management of drawdowns and overall stability. However, both the original analysis and the additional reproduction reported in this version show that ensemble performance is sensitive to the choice of variance threshold \(\tau\), classifier group, RL-agent pair, and market universe. The reproduction evidence strengthens the conclusion that classifier-assisted ensemble selection can improve robustness, while also clarifying that the advantage is conditional rather than automatic across all datasets. This study emphasizes the value of combining RL with classifiers for adaptive decision-making, with implications for financial trading, robotics, and other dynamic environments.

This paper develops an asymptotic and inferential theory for fixed-effects panel quantile regression (FEQR) that delivers inference robust to pervasive common shocks. Such shocks induce cross-sectional dependence that is central in many economic and financial panels but largely ignored in existing FEQR theory, which typically assumes cross-sectional independence and requires $T \gg N$. We show that the standard FEQR estimator remains asymptotically normal under the mild condition $(\log N)^2/T \to 0$, thereby accommodating empirically relevant regimes, including those with $T \ll N$. We further show that common shocks fundamentally alter the asymptotic covariance structure, rendering conventional covariance estimators inconsistent, and we propose a simple covariance estimator that remains consistent both in the presence and absence of common shocks. The proposed procedure therefore provides valid robust inference without requiring prior knowledge of the dependence structure, substantially expanding the applicability of FEQR methods in realistic panel data settings.

Renewable deployment and rising demand from electrification and large digital loads are transforming electricity markets. However, how these developments reshape electricity price dynamics remains poorly understood, leaving system planners, capacity investors, and market participants reliant on assumptions from a thermal-dominated era that may no longer hold. We use causal discovery to study the evolution of wholesale electricity prices in Texas, which is undergoing rapid transformation. Our findings overturn the view of Texas as a gas-price-driven market, demonstrating that wind generation has become the dominant causal driver of day-ahead prices, with effects more than three times greater than those of natural gas. Yet wind's price-suppressing effect is weakening during peak periods, and wind growth redistributes congestion costs to distant load centres. Furthermore, rising load in South and West Texas alters system prices and regional differentials. Uncovering the evolving spatiotemporal nature of causal drivers, our analysis reveals that the pace, geographic siting, and relative scale of new generation and large loads will be decisive for future electricity price risks, infrastructure needs, and investments.

AI-assisted task delegation is increasingly common, yet human effort in such systems is costly and typically unobserved. Recent work by Bastani and Cachon (2025); Sambasivan et al. (2021) shows that accuracy-based payment schemes suffer from incentive collapse: as AI accuracy improves, sustaining positive human effort requires unbounded payments. We study this phenomenon in a budget-constrained principal-agent framework with strategic human agents whose output accuracy depends on unobserved effort. Our first contribution is a general impossibility result showing that incentive collapse is not merely a limitation of simple linear payments, but arises for any payment rule based only on observed task this http URL overcome this barrier, we propose a sentinel-auditing payment mechanism that enforces a strictly positive and controllable level of human effort at finite cost, independent of AI accuracy. Building on this incentive-robust foundation, we develop an incentive-aware active statistical inference framework that jointly optimizes (i) the auditing rate and (ii) active sampling and budget allocation across tasks of varying difficulty to minimize the final statistical loss under a single budget. Experiments demonstrate improved cost-error tradeoffs relative to standard active learning and auditing-only baselines.

We study the privacy of releasing posterior sample paths from a Gaussian process (GP) when the entire training set including covariates and responses is private. Unlike standard differential-privacy (DP) mechanisms that add external noise, posterior sampling is random by construction. We show that this intrinsic randomness yields DP guarantees by deriving explicit Rényi-DP bounds for GP posterior sample-path release. The bounds separate posterior-mean leakage from data-dependent posterior-covariance leakage showing that meaningful privacy depends sharply on effective ridge regularisation. We apply membership-inference attacks to show that empirical leakage follows the predicted dependence on regularisation, posterior variance and the number of released posterior sample-paths. Utility experiments on downstream posterior-sampling tasks identify noisy-observation regimes where privacy-compatible regularisation preserves useful decisions with modest utility loss. When stronger privacy is needed, the intrinsic guarantee can be sharpened by adding calibrated GP noise, providing an explicit additional privacy knob.

Graph-based learning methods have become increasingly prominent due to their strong performance across diverse applications. Among these, recent frameworks grounded in diffusion processes provide a unifying perspective that extends traditional graph neural network formulations while addressing limitations of standard message-passing mechanisms. Despite these advances, concerns remain regarding the fairness of such models, as they may propagate or amplify biases present in the data. In this work, we introduce a fairness-aware adaptation of graph-based diffusion by modifying the underlying Laplacian operator. Our approach incorporates multiple complementary transformations, including subspace projections, spectral adjustments, and frequency-based filtering, to mitigate bias-related components. Leveraging the intrinsic smoothing properties of graph diffusion, we provide a principled analysis of the resulting behavior and establish theoretical insights into fairness properties. We evaluate the proposed framework on both synthetic and real-world datasets, demonstrating that it achieves competitive performance while improving fairness metrics with limited additional computational cost.

The rapid deployment of AI systems in high-stakes domains, including those classified as high-risk under the The EU AI Act (Regulation (EU) 2024/1689), has intensified the need for reliable compliance auditing. For binary classifiers, regulatory risk assessment often relies on global fairness metrics such as the Disparate Impact ratio, widely used to evaluate potential discrimination. In typical auditing settings, the auditee provides a subset of its dataset to an auditor, while a supervisory authority may verify whether this subset is representative of the full underlying distribution. In this work, we investigate to what extent a malicious auditee can construct a fairness-compliant yet representative-looking sample from a non-compliant original distribution, thereby creating an illusion of fairness. We formalize this problem as a constrained distributional projection task and introduce mathematically grounded manipulation strategies based on entropic and optimal transport projections. These constructions characterize the minimal distributional shift required to satisfy fairness constraints. To counter such attacks, we formalize representativeness through distributional distance based statistical tests and systematically evaluate their ability to detect manipulated samples. Our analysis highlights the conditions under which fairness manipulation can remain statistically undetected and provides practical guidelines for strengthening supervisory verification. We validate our theoretical findings through experiments on standard tabular datasets for bias detection. Code is publicly available at this https URL.

Credal predictors are models that are aware of epistemic uncertainty and produce a convex set of probabilistic predictions. They offer a principled way to quantify predictive epistemic uncertainty (EU) and have been shown to improve model robustness in various settings. However, most state-of-the-art methods mainly define EU as disagreement caused by random training initializations, which mostly reflects sensitivity to optimization randomness rather than uncertainty from deeper sources. To address this, we define EU as disagreement among models trained with varying relaxations of the i.i.d. assumption between training and test data. Based on this idea, we propose CreDRO, which learns an ensemble of plausible models through distributionally robust optimization. As a result, CreDRO captures EU not only from training randomness but also from meaningful disagreement due to potential distribution shifts between training and test data. Empirical results show that CreDRO consistently outperforms existing credal methods on tasks such as out-of-distribution detection across multiple benchmarks and selective classification in medical applications.

We address the problem of training conversion prediction models in advertising domains under privacy constraints, where direct links between ad clicks and conversions are unavailable. Motivated by privacy-preserving browser APIs and the deprecation of third-party cookies, we study a setting where the learner observes a sequence of clicks and a sequence of conversions, but can only link a conversion to a set of candidate clicks (an attribution set) rather than a unique source. We formalize this as learning from attribution sets generated by an oblivious adversary equipped with a prior distribution over the candidates. Despite the lack of explicit labels, we construct an unbiased estimator of the population loss from these coarse signals via a novel approach. Leveraging this estimator, we show that Empirical Risk Minimization achieves generalization guarantees that scale with the informativeness of the prior and is also robust against estimation errors in the prior, despite complex dependencies among attribution sets. Simple empirical evaluations on standard datasets suggest our unbiased approach significantly outperforms common industry heuristics, particularly in regimes where attribution sets are large or overlapping.

When deploying a single predictor across multiple subpopulations, we propose a fundamentally different approach: interpreting group fairness as a bargaining problem among subpopulations. This game-theoretic perspective reveals that existing robust optimization methods such as minimizing worst-group loss or regret correspond to classical bargaining solutions and embody different fairness principles. We propose relative improvement, the ratio of actual risk reduction to potential reduction from a baseline predictor, which recovers the Kalai-Smorodinsky solution. Unlike absolute-scale methods that may not be comparable when groups have different potential predictability, relative improvement provides axiomatic justification including scale invariance and individual monotonicity. We establish finite-sample convergence guarantees under mild conditions.

Current practices for reporting differential privacy (DP) guarantees for machine learning (ML) algorithms such as DP-SGD provide an incomplete and potentially misleading picture. For instance, if only a single $(\varepsilon, \delta)$ is known about a mechanism, standard analyses show that there could exist highly accurate inference attacks against training data records, when, upon a more careful analysis, such accurate attacks do not exist for most practical mechanisms. In this position paper, we argue that using _non-asymptotic_ Gaussian Differential Privacy (GDP) as the primary means of communicating DP guarantees in ML avoids these potential downsides. Using two recent developments in the DP literature: (i) open-source numerical accountants capable of computing the privacy profile and $f$-DP curves of DP-SGD to arbitrary accuracy, and (ii) a decision-theoretic metric over DP representations, we show how to provide non-asymptotic bounds on GDP using numerical accountants, and show that GDP can capture the entire privacy profile of DP-SGD and related algorithms with virtually no error, as quantified by the metric. To support our claims, we investigate the privacy profiles of state-of-the-art DP large-scale image classification, and the TopDown algorithm for the U.S. Decennial Census, observing that GDP fits their profiles remarkably well in all cases. We conclude with a discussion on the strengths and weaknesses of this approach, and discuss which other privacy mechanisms could benefit from GDP.

Prior to the version 1.3.1 update on CRAN in December 2025, gsynth, a popular R package for estimating Interactive Fixed Effects (IFE) models, could drastically and systematically underestimate standard errors. This underestimation would occur when two estimation options (inference = "parametric", and EM = TRUE) were used together, in which case the package would apply a parametric bootstrap procedure to Gobillon and Magnac (2016)'s IFE-EM estimator. The package ceased supporting this combination in December 2025, and the latest documentation now describes the parametric bootstrap as not suitable for use with the IFE-EM estimator due to a theoretical incompatibility. Our focus is an implementation error we identified in the pre-1.3.1 versions of gsynth: the parametric bootstrap used when EM = TRUE did not match the algorithm proposed in Xu (2017), using in-sample residuals instead of out-of-sample errors. We show that this implementation error alone can cause underestimation by orders of magnitude. We conduct an empirical Monte Carlo study using randomly assigned placebo treatments on a series of state-level panel datasets, and show that gsynth could yield high false positive rates in realistic settings. We identify three papers published in the American Political Science Review that are affected by this behavior. Reanalyzing the relevant sections of these papers, we show that (i) correcting the implementation error renders most findings insignificant, and (ii) using Xu (2017)'s Generalized Synthetic Control method in place of IFE-EM renders every finding insignificant.

In the perpetual pursuit of performance, modern computing systems rely ever more on stateful mechanisms to accommodate the dynamics of workloads and physical environments, bolstering efficiency but confounding benchmarking and thereby the optimization of software. Indeed, by their nature, adaptive mechanisms introduce temporal dependencies between measurements and render naive estimators of individual program performance biased. Observing that rectifying such biases necessitates speculative assumptions about system dynamics, we call for prioritizing performance differentials over absolute measures and formalize software benchmarking as the decision problem of identifying the fastest program, for which relative knowledge suffices. To this end, we propose simple experiment designs admitting consistent estimators of contrasts, whereby program-specific biases cancel under tenable assumptions. These designs asymptotically yield the correct decision and afford a robust methodology for finite-budget benchmarking in stateful environments, bearing broad implications for the development of performance-sensitive software.

Remote sensing techniques have been increasingly utilised in aquatic applications in recent years. A common challenge in using optical satellite data is the presence of missing observations due to cloud cover. These data gaps can lead to missed detection of critical events, such as algal blooms, in lakes of high interest to water authorities. As a result, enhancing the completeness of optical satellite datasets is crucial for improving the monitoring and prediction of algal blooms. In this study, we compared a traditional data imputation method (i.e., linear interpolation) with deep learning models for reconstructing missing spectral bands across four lakes with historical records of algal blooms. The deep learning models adopted include CNN-based architectures (i.e., CNN, Inception Resnet, and Autoencoder) and CNN-LSTM-based architectures (i.e., CNN-LSTM, Resnet-LSTM, and Autoencoder-LSTM). Our results demonstrated that deep learning models substantially outperformed the baseline linear interpolation method in imputing spectral band values within artificially masked regions. Among these models, CNN delivered the best performance across most lakes. Furthermore, we evaluated the performance of algal bloom indices (i.e., Green/Red and NDCI) derived from the imputed imagery by comparing them with the observed data. Our results demonstrate that deep learning models are effective for imputing missing data in PlanetScope SuperDove imagery, enabling more reliable applications in water monitoring.

Statistical methods that leverage external trial information to help accelerate drug development are becoming increasingly popular. Bayesian methods facilitate dynamic borrowing, where the similarity of the response guides how much information is used. We have proposed a semiparametric Bayesian borrowing model for time-to-event data, with smoothing priors that allows the baseline hazard to take any form via an ensemble average \citep{Scott2024}. By accurately modelling the baseline hazard, rather than approximating its form via fixed piecewise intervals, power is improved and bias of the estimated treatment effect reduced when the borrowing assumption of parameter exchangeability holds. A ``lump-and-smear'' borrowing prior makes the model robust to non-exchangeable historical data by increasing the sensitivity of borrowing to the presence of prior-data conflict, reducing the potential for type I error inflation. We present BayesFBHborrow, an R package that implements our semiparametric Bayesian borrowing model with a historical control. We demonstrate how to select the optimal borrowing hyperparameters. The model supports covariate-adjusted borrowing, which can reduce prior-data conflict and improve power when differences in outcomes are attributable to changes in the covariate distribution. As the treatment effect estimator is non-collapsible, the marginal hazard ratio can be estimated via Bayesian G-computation, while still permitting an adjusted analysis to account for control group drift. We illustrate the Bayesian flexible baseline hazard model on a simulated and real dataset with a marginal estimand, for both an unadjusted and adjusted analyses.

We present a framework to compute non-Gaussian likelihoods for two-point correlation functions. The non-Gaussianity is most pronounced on large scales that will be well-measured by stage-IV weak-lensing surveys. We show how such a multivariate likelihood can be constructed and efficiently evaluated using a copula approach by incorporating exact one-dimensional marginals and a dependence structure derived from the exact multivariate likelihood. The copula likelihood is found to be in better agreement with simulated sampling distributions of correlation functions than Gaussian likelihoods, particularly on large scales. We furthermore investigate the effect of the non-Gaussian copula likelihood on posterior inference, including sampling the full parameter space of contemporary weak-lensing analyses. We find potential parameter shifts in $S_8$ on the order of one standard deviation for $1 \ 000 \ \mathrm{deg}^2$ surveys but negligible shifts for areas of $10 \ 000 \ \mathrm{deg}^2$, suggesting Gaussian likelihoods are sufficient for stage-IV surveys, though results depend on the detailed mask geometry and data-vector structure.

Wireframe parsing aims to recover line segments and their junctions to form a structured geometric representation useful for downstream tasks such as Simultaneous Localization and Mapping (SLAM). Existing methods predict lines and junctions separately and reconcile them post-hoc, causing mismatches and reduced robustness. We present Co-PLNet, a point-line collaborative framework that exchanges spatial cues between the two tasks, where early detections are converted into spatial prompts via a Point-Line Prompt Encoder (PLP-Encoder), which encodes geometric attributes into compact and spatially aligned maps. A Cross-Guidance Line Decoder (CGL-Decoder) then refines predictions with sparse attention conditioned on complementary prompts, enforcing point-line consistency and efficiency. Experiments on Wireframe and YorkUrban show consistent improvements in accuracy and robustness, together with favorable real-time efficiency, demonstrating our effectiveness for structured geometry perception. Our code is available at this https URL.