Geostatistics aims to infer a spatially continuous phenomenon from observations collected at a finite number of locations, frequently measured with error. Whenever there is stochastic dependence between the spatial and sampling processes, preferential sampling occurs. Ignoring this problem drives to incorrect and biased estimates and, therefore, recognizing it is quite important, but not always simple to execute and understand. In this work, a test for assessing preferential sampling, simple and easy to implement, is presented, overcoming the previous concerns. It is based on the dependence between the number of sampled points and the values of the corresponding measures. The performance of the proposed test id assessed through a large simulation study, which consideres different levels of preferentiability, relation with a covariate, different sample sizes and different test procedure conditions. The results are quite encouraging, with high levels of correct preferential sampling detections, further confirmed by the test application to already known real data sets of lead concentrations in moss samples and red and blue shrimp capture data.

Ridge-regularized Hotelling-type (RHT) change-point tests depend on a ridge parameter $λ$, but the power-optimal value is determined by the unknown covariance structure and the unknown mean shift. We avoid selecting a single ridge value by computing fixed-ridge p-values on a finite deterministic grid and aggregating them with the Cauchy combination rule. Under the standard random-matrix conditions for fixed-ridge RHT statistics, we establish finite-grid joint weak convergence of the ridge processes. This leads to fixed-level validity under joint-limit calibration and small-tail validity for the analytic Cauchy p-value. Monte Carlo experiments show that deterministic-grid Cauchy aggregation has stable size behavior and achieves power close to the best stable fixed ridge choice across a range of covariance and signal configurations.

The tail index parameter of heavy-tailed probability models plays a key role in characterizing the tail decay of the underlying distribution function and is often involved in extrapolation procedures for various extreme value analysis questions. In this paper we revisit the question of tail index estimation and combine the ideas of bias-correction and empirical likelihood estimation to propose an estimator that offers an attractive alternative to some of the existing estimators. We develop an asymptotic theory for the proposed estimator and conduct simulation studies to demonstrate its performance in finite sample situations. The method is also applied to a data example for illustration.

This paper develops a unified framework for the study of real-order moments of arbitrary random variables. General integral representations are established in terms of cumulative distribution functions and survival functions, covering continuous, discrete, and mixed distributions supported on the whole real line. These formulas extend the classical tail-integral identities for nonnegative random variables and provide a common treatment of positive, fractional, and negative moments. For discrete distributions, explicit series representations are derived in terms of cumulative probabilities, yielding simple criteria for the existence of moments. Applications are presented for the zeta and Skellam distributions, illustrating how tail behavior determines moment finiteness and how moments can be represented geometrically through cumulative distribution functions. In addition, a representation for logarithmic moments is obtained, linking logarithmic means, Laplace transforms, and the classical Frullani identity. The results provide a unified perspective on moment representations and establish useful connections between tail probabilities, distribution functions, Laplace transforms, and moment existence.

We develop a scan statistic method for detecting local clusters in a two-sample nonhomogeneous Poisson process (NHPP) framework, motivated by copy number variation (CNV) analysis in next-generation sequencing data. The control sample is used to construct an empirical time transformation, under which the transformed case sample is approximately uniform on [0,1] under the null hypothesis. The scan statistic is defined as the maximum number of transformed points within a moving window. We show that the scan statistic converges to a generalized extreme value (GEV) distribution with an extremal index that captures the dependence induced by overlapping windows. The GEV parameters and extremal index are estimated using maximum likelihood and exceedance clustering methods, providing an asymptotic calibration of the test. A permutation procedure is also developed to provide a nonparametric alternative. Simulation studies show that the permutation calibration maintains empirical Type I error close to the nominal level across the considered settings, and the GEV calibration is accurate for smaller windows. Both proposed procedures show competitive power compared with the continuous testing method under heterogeneous baseline intensities. An application to sequencing data illustrates the effectiveness of the proposed approach for detecting CNV regions.

We study the finite-sample behavior of the Generalized Fisher Transformation (GFT), the parametrization of a correlation matrix $C$ by $γ(C)=\operatorname{vecl}\log C$. The GFT coordinates extend Fisher's transformation to dimension $n>2$: for elliptical data their finite-sample distributions are close to Gaussian. More strikingly, the coordinates are nearly uncorrelated and their covariance is largely invariant to $C$. This approximate orthogonality and invariance make GFT-based inference far better behaved in finite samples than inference based on sample correlations or element-wise Fisher transformed correlations, yielding estimation errors that are approximately Gaussian, weakly dependent, and nearly pivotal.

Machine-learned anomaly detection is reshaping searches for new physics, but it has outrun the statistics used to interpret it. A raw anomaly score has no calibrated meaning, a model that scans many regions inflates the look-elsewhere effect, and the asymptotic significances the field relies on are blind to the background mismodelling that anomaly detectors are especially prone to. We propose a calibration layer, built on conformal prediction, that turns any anomaly score into a defensible significance with distribution-free, finite-sample guarantees. Conformal prediction converts scores into valid local p-values, weighted and Mondrian variants repair the sideband-to-signal-region exchangeability failures that resonant searches suffer, and a Gross-Vitells step carries the result through to a look-elsewhere-aware global significance. The layer does two things at once. It exposes miscalibration that the standard pipeline cannot see, and it corrects it without retraining the detector. On public LHC Olympics data, a classifier develops a substructure-mass correlation that makes sideband-calibrated background p-values anti-conservative. Taken at face value, this manufactures a $\sim 46σ$ excess from background sculpting alone, which the label-free weighted correction removes, restoring an honest null. When run as a blind wide-mass bump hunt, the standard asymptotic and unweighted procedures fabricate $\gtrsim10σ$ excesses and $\approx5σ$ excesses even in signal-free windows, while the conformal layer raises no false alarms and its global false-positive rate is verified on background-only pseudoexperiments. The result is an auditable, detector-agnostic path from an uncalibrated score to a trials-factor-aware significance, ready to be folded into experimental anomaly searches.

Kling-Gupta efficiency ($\mathrm{KGE}$) is a model performance evaluation metric widely used in hydrology, but its properties as a statistical estimator have remained unexplored. We formalize the Kling-Gupta loss $L_\mathrm{KG} = (1 - \mathrm{KGE})^2$ in an extremum estimation framework (maximizing $\mathrm{KGE}$) for multiple linear regression. We give explicit formulas showing that Kling-Gupta regression scales the ordinary least squares (OLS) coefficient vector by a variance-inflation factor depending on sample variances and covariances. Its predictions reproduce the training set response variance, unlike OLS's variance reduction, while both maintain the response mean and achieve the same sample correlation. We prove that no estimator simultaneously maximizes Nash-Sutcliffe efficiency ($\mathrm{NSE}$) and $\mathrm{KGE}$: OLS maximizes $\mathrm{NSE}$ but not $\mathrm{KGE}$, whereas Kling-Gupta regression maximizes $\mathrm{KGE}$ at the expense of $\mathrm{NSE}$. We establish almost-sure convergence of the Kling-Gupta estimator to well-defined population limits. The training and test set performance metrics for both estimators converge asymptotically to identical limits (different for OLS vs. Kling-Gupta). In a single-predictor model with fixed intercept, we identify conditions where a global minimum of $L_\mathrm{KG}$ does not exist because of discontinuity at zero slope. This work establishes a mathematical foundation for $\mathrm{KGE}$-based estimation and clarifies its effects on predictive performance in hydrologic modeling.

We investigate the behavior of the length of the longest weakly increasing subsequences (weak LIS) of $n$-step random walks with nonzero integer increments $k = \pm 1, \pm 2, \dots$ given by a symmetric heavy tailed mass distribution proportional to $|k|^{-1-α}$ for several values of the real parameter $α> 0$ together with that of the simple random walk ($k=\pm 1$), to which the $n$-step heavy tailed walks reduce when $α$ grows large enough that step jumps beyond $\pm 1$ become essentially absent on the scale of $n$. By means of exploratory fits, weighted nonlinear least squares, and nested-model comparisons, we found that the sample average length $\langle{L_{n}}\rangle$ scales like $\langle{L_{n}}\rangle \sim \sqrt{n}\log{n}$ when the distribution of increments has finite variance ($α> 2$) and $\langle{L_{n}}\rangle \sim n^θ$ with a varying exponent $θ> 0.5$ when the variance is infinite ($α\leq 2$). Distributional diagnostics indicate that the bulk of the $L_{n}$ distribution is very well-approximated by a lognormal model, though systematic deviations are observed in the tails. Our results corroborate and expand upon previous results for the LIS of other types of heavy-tailed random walks and raise a conjecture as to whether the distribution of $L_{n}$ is given, or can be effectively described, by a lognormal distribution.

We characterize structures such as monotonicity, convexity, and modality in smooth regression curves using persistent homology. Persistent homology is a key tool in topological data analysis that detects higher-dimensional topological features such as connected components and holes (cycles or loops) in the data. In other words, persistent homology is a multiscale version of homology that characterizes sets based on the connected components and holes. We use super-level sets of functions to extract geometric features via persistent homology. In particular, we explore structures in regression curves via the persistent homology of super-level sets of a function, where the function of interest is - the first derivative of the regression function. In the course of this study, we extend an existing procedure of estimating the persistent homology for the first derivative of a regression function and establish its consistency. Moreover, as an application of the proposed methodology, we demonstrate that the persistent homology of the derivative of a function can reveal hidden structures in the function that are not visible from the persistent homology of the function itself. In particular, we characterize structures such as monotonicity, convexity, and modality, and propose a measure of statistical significance to infer these structures in practice. Finally, we conduct an empirical study to implement the proposed methodology on simulated and real data sets and compare the derived results with an existing methodology.

Valid uncertainty quantification for subsampling-based and randomized estimators often depends on variance estimators whose behavior is much less understood than that of the underlying point estimator. We prove ratio-consistency of the jackknife variance estimator, and certain delete-$d$ variants, for a broad class of generalized U-statistics whose variance is asymptotically dominated by their Hajek projection and whose normalized first-projection squares satisfy a row-wise $L^r$ weak law, with the classical fixed-order case recovered as a special instance. This projection-dominance plus square-LLN structure unifies and generalizes several criteria from the existing literature, clarifies when the simple nonparametric jackknife is theoretically justified in the generalized setting, and yields consistent variance estimation for the two-scale distributional nearest-neighbor regression estimator under substantially weaker conditions than previously required.

The predictive approach to Bayesian inference accesses the posterior distribution via a sequence of one-step-ahead predictives, enabling inference via predictive resampling without Markov chain Monte Carlo. In the random-design regression setting, an explicit specification of the predictive design distribution is required, yet the impact of this choice has received little formal attention. We study the role of this predictive design distribution in parametric martingale posteriors for regression, and identify predictive notions of identifiability and design invariance that are essential for valid inference, particularly in the high-dimensional regression setting. Building on these foundations, we introduce a novel class of parametric martingale posteriors for regression that satisfies a weak form of these desiderata, and naturally accommodates the high-dimensional setting through regularization. We then illustrate our method through a simulation.

We discuss probabilistic measures of concordance between two probability distributions based on the expected misclassification rate (EMR). The focus is on comparing a given reference distribution with other distributions in a parametrised class, and optimising concordance by identifying parameter values maximising EMR or a regularised variant. EMR is a practical and decision-theoretically meaningful measure, and its optimisation has direct interpretation as a Bayesian decision analysis with a bounded utility function. We explore theoretical properties of EMR, discuss relationships with other measures including Küllback-Leibler divergence, and recognise that its optimisation has a synthetic Bayesian emulation interpretation that aids understanding and specification of regularisation penalties. A main area of methodology is in mixture synthesis where the parametrised family is a discrete mixture of given distributions. A detailed example comes from scenario forecasting in macroeconomic policy settings, a key applied area motivating the new methodology. Theoretical developments underlie efficient numerical optimisation and analysis is easily implemented using direct Monte Carlo simulation.

Bayesian statistics has gained popularity in psychological research due to its intuitive uncertainty quantification and convenient information-updating rules. In many applications, however, prior distributions are introduced merely as instruments to facilitate computation, rather than as representations of genuine subjective belief. Consequently, relying on standard Bayesian justifications for inferential procedures becomes conceptually ungrounded. In this paper, we recommend evaluating finite-sample performance over repeated sampling of data and parameters as an alternative justification for "pragmatic Bayes." We demonstrate a key vulnerability in the usual posterior-based inference: when analysts' chosen prior distribution mismatches the true parameter-generating process, Bayesian inference can be misleading. Given that this true process is rarely known in practice, we propose a safer alternative: calibrating Bayesian credible regions to achieve frequentist validity. This latter criterion is stronger and guarantees validity of Bayesian inference regardless of the underlying parameter-generating mechanism. To solve the calibration problem in practice, we propose a novel stochastic approximation algorithm. A Monte Carlo experiment is conducted and reported, in which we observe that uncalibrated Bayesian inference can be liberal under certain parameter-generating scenarios, whereas our calibrated solution consistently maintain validity. We also illustrate the proposed calibration procedure using a real-data example involving location-scale regression.

Computation in biological systems is fundamentally energy-constrained, yet standard theories of computation treat energy as freely available. Here, we argue that variational free energy minimization under a Poisson assumption offers a principled path toward an energy-aware theory of computation. Our key observation is that the Kullback-Leibler (KL) divergence term in the Poisson free energy objective becomes proportional to the prior firing rates of model neurons, yielding an emergent metabolic cost term that penalizes high baseline activity. This structure couples an abstract information-theoretic quantity -- the *coding rate* -- to a concrete biophysical variable -- the *firing rate* -- which enables a trade-off between coding fidelity and energy expenditure. Such a coupling arises naturally in the Poisson variational autoencoder (P-VAE) -- a brain-inspired generative model that encodes inputs as discrete spike counts and recovers a spiking form of *sparse coding* as a special case -- but is absent from standard Gaussian VAEs. To demonstrate that this metabolic cost structure is unique to the Poisson formulation, we compare the P-VAE against Grelu-VAE, a Gaussian VAE with ReLU rectification applied to latent samples, which controls for the non-negativity constraint. Across a systematic sweep of the KL term weighting coefficient $β$ and latent dimensionality, we find that increasing $β$ monotonically increases sparsity and reduces average spiking activity in the P-VAE. In contrast, Grelu-VAE representations remain unchanged, confirming that the effect is specific to Poisson statistics rather than a byproduct of non-negative representations. These results establish Poisson variational inference as a promising foundation for a resource-constrained theory of computation.

Simulation-based inference (SBI) enables amortized Bayesian inference by first training a neural posterior estimator (NPE) on prior-simulator pairs, typically through low-dimensional summary statistics, which can then be cheaply reused for fast inference by querying it on new test observations. Because NPE is estimated under the training data distribution, it is susceptible to misspecification when observations deviate from the training distribution. Many robust SBI approaches address this by modifying NPE training or introducing error models, coupling robustness to the inference network and compromising amortization and modularity. We introduce minimum-distance summaries, a plug-in robust NPE method that adapts queried test-time summaries independently of the pretrained NPE. Leveraging the maximum mean discrepancy (MMD) as a distance between observed data and a summary-conditional predictive distribution, the adapted summary inherits strong robustness properties from the MMD. We demonstrate that the algorithm can be implemented efficiently with random Fourier feature approximations, yielding a lightweight, model-free test-time adaptation procedure. We provide theoretical guarantees for the robustness of our algorithm and empirically evaluate it on a range of synthetic and real-world tasks, demonstrating substantial robustness gains with minimal additional overhead.

Penalized transformation models (PTMs) are a semiparametric location-scale regression family that estimate a response's conditional distribution directly from the data, and model the location and scale through structured additive predictors. The core of the model is a monotonically increasing transformation function that relates the response distribution to a reference distribution. The transformation function is equipped with a smoothness prior that regularizes how much the estimated distribution diverges from the reference. PTMs can be seen as a bridge between conditional transformation models and generalized additive models for location, scale and shape. Markov chain Monte Carlo inference for PTMs offers straightforward uncertainty quantification for the conditional distribution as well as for the covariate effects. A simulation study demonstrates the effectiveness of the approach and includes comparisons to many alternative methods. Applications to the Fourth Dutch Growth Study and the Framingham Heart Study illustrate the usage and practical utility. A full-featured implementation is available as a Python library. Supplementary material for this article is available online.

Valid treatment effect inference in survival studies is fundamental yet challenging when the treatment assignments and outcomes are confounded by many baseline covariates. To this end, in this paper we propose a high-dimensional survival causal inference (HSCI) framework that delivers valid inference under a sparse high-dimensional Cox proportional hazards outcome model and a high-dimensional logistic propensity score working model. To mitigate the nuisance estimation bias, we develop a Neyman near-orthogonal score for the treatment effect and implement it with cross-fitting. Under doubly robust nuisance-rate conditions, we establish the root-n asymptotic normality and consistent variance estimation. We also extend the framework to inference on high-dimensional survival covariate effects. Simulation examples confirm that HSCI reduces sharply the bias relative to the regularized Cox estimators and maintains valid confidence interval coverage across different dimensionality, censoring, and misspecified propensity-model settings. An application to diffuse large-B-cell lymphoma data further showcases its value for high-dimensional biomedical survival studies.

Observational studies frequently involve hierarchical data structures in which individuals are nested within higher-level units. In such settings, unmeasured cluster-level factors may confound the treatment-outcome relationship and may additionally induce treatment effect heterogeneity across clusters, complicating causal effect estimation. We formalize the use of g-computation for hierarchical observational data by incorporating random-effects models (REM) as outcome models and propose a within-group g-computation strategy designed to reduce bias arising from unmeasured cluster context. The approach groups clusters according to their observed treatment prevalence and performs g-computation within groups before aggregating group-specific estimates. Through extensive Monte Carlo simulations, we compare the standard and within-group implementations of g-computation using both linear models and REM. Results show that both standard and within-group REM-based implementations substantially reduce bias when the unmeasured cluster-level variable acts solely as a confounder, whereas the proposed within-group REM estimator achieves the lowest RMSE when the unmeasured cluster-level factor acts as both a confounder and a source of treatment effect heterogeneity. We apply the proposed within-group REM estimator to estimate the causal effect of adolescent pregnancy on the child height-for-age Z-score using 2019 Bangladesh MICS data, obtaining an estimated effect of -0.12 (95% bootstrap CI: [-0.18, -0.06]). The proposed within-group g-computation framework offers a strategy for reducing bias from unmeasured cluster-level confounding and treatment effect heterogeneity in hierarchical observational studies.

Comparing causal relationships across populations is essential in many scientific domains. This paper studies the problem of inferring a difference graph between two environments and proposes a causal discovery method for linear structural causal models based on equality tests of regression coefficients. We show that invariance of regression coefficients is governed by graphical conditions that go beyond standard d-separation. Therefore, we introduce diff-separation, a graphical criterion that characterizes when a conditioning set blocks all paths capable of inducing differences in regression coefficients across environments. Building on this criterion, we introduce a corresponding diff-faithfulness assumption, linking graphical diff-separation statements to equality constraints on regression coefficients. Finally, we propose LDiffPC, a PC-style algorithm that uses equality tests of regression coefficients to recover the differences from multi-environment data.

Causal inference typically assumes centralized access to individual-level data. Yet, in practice, data are often decentralized across multiple sites, making centralization infeasible due to privacy, logistical, or legal constraints. We address this problem by estimating the Average Treatment Effect (ATE) from decentralized observational data via a Federated Learning (FL) approach, allowing inference through the exchange of aggregate statistics rather than individual-level data. We propose a novel method to estimate propensity scores via a federated weighted average of local scores using Membership Weights (MW), defined as probabilities of site membership conditional on covariates. MW can be flexibly estimated with parametric or non-parametric classification models using standard FL algorithms. The resulting propensity scores are used to construct Federated Inverse Propensity Weighting (Fed-IPW) and Augmented IPW (Fed-AIPW) estimators. In contrast to meta-analysis methods, which fail when any site violates positivity, our approach exploits heterogeneity in treatment assignment across sites to improve overlap. We show that Fed-IPW and Fed-AIPW perform well under site-level heterogeneity in sample sizes, treatment mechanisms, and covariate distributions. Theoretical analysis and experiments on simulated and real-world data demonstrate clear advantages over meta-analysis and related approaches.

We introduce a new convex optimization framework for logistic scalar-on-matrix regression which incorporates nuclear and $\ell_1$ norm penalties to enforce simultaneously low-rank and sparse structures in the estimated coefficient matrix. The proposed method enables interpretable modeling of high-dimensional matrix-valued predictors in the presence of binary responses. We derive a custom algorithm based on the Alternating Direction Method of Multipliers (ADMM) to efficiently solve the resulting convex optimization problem and establish the theoretical properties of the obtained solution. Numerical experiments clearly demonstrate the effectiveness of our method in recovering meaningful predictive patterns. Finally, we apply our method to the brain imaging data to identify structures in functional brain connectivity matrices that are characteristic of subjects with a family history of alcohol use disorders (AUDs).

We propose a generalized debiased Lasso estimator based on a stability principle. When a single column of the design matrix is perturbed, the estimator admits a simple update formula that can be computed from the original solution. Under sub-Gaussian designs with well-conditioned covariance, this approximation is asymptotically accurate for all but a vanishing fraction of coordinates in the proportional growth regime. The proof relies on concentration and anti-concentration arguments to control error terms and sign changes. In contrast, establishing comparable distributional limits (e.g., Gaussianity) under similar assumptions remains open. As an application, we show that the approximation significantly reduces the computational cost of resampling-based variable selection procedures, including the conditional randomization test and a local knockoff filter.

In this article, we connect statistical inference for spatial point processes with the analysis of waiting pedestrian crowds through two interconnected contributions. First, on the methodological side we develop an inference procedure for semiparametric spatial point process models leveraging replicated spatial patterns, i.e., multiple approximately independent realizations from the same process. Second, we show that spatial point processes provide a suitable modeling framework for waiting pedestrians, capturing two key aspects: spatial inhomogeneity driven by location attractiveness and repulsive interactions between pedestrians. These two components are central to the inference problem itself, since spatial point process modeling hinges on disentangling background intensity from interaction. Although replicated spatial patterns are rare in point process literature, they are available here through a unique real-life pedestrian dataset, thereby directly linking the methodological development to the physical application. We use the proposed methods to fit and evaluate determinantal and Gibbs point processes in a simulation study and a real-world case study. Despite persistent challenges in decoupling the influences of inhomogeneity from interaction, these models are able to reproduce key empirical features of waiting pedestrians.

High\text{--}resolution scientific data, such as geomagnetic index streams, often exhibit complex temporal dependencies that can be modeled through functional data analysis. Conventional functional time series (FTS) methods typically partition continuous processes into non-overlapping segments, which artificially fragments temporal continuity and can limit estimation efficiency and stability. This is particularly evident in geomagnetic time series prediction due to their noisy, sudden, and large\text{--}scale changes. This study presents a robust multivariate FTS forecasting framework for multi\text{--}dimensional time series with inter\text{--}series correlations and the existence of exogenous predictors. We introduce an overlapping rolling\text{--}window scheme that preserves temporal coherence and mitigates boundary information loss, thereby enriching the effective sample size for a more efficient and stable estimation. We integrate functional principal component analysis for dimension reduction with a vector autoregressive model with exogenous inputs to capture latent dynamics across correlated series. We also construct computationally efficient conformal prediction intervals for uncertainty quantification. The framework is motivated by and applied to the simultaneous forecasting of five critical geomagnetic indices, Kp, Dst, SYM\text{--}H, SME, and SMR, using solar wind parameters as predictors. Empirical results show that this approach outperforms state\text{--}of\text{--}the\text{--}art machine learning baselines, extends forecast horizons to 6\text{--}24 hours, and provides calibrated uncertainty bounds.

Real-world spatio-temporal forecasting must handle irregular time points, spatially sparse observations, and the need for uncertainty quantification. This setting is often further compounded by nonlocal interactions (long-range spatial coupling). Modeling continuous-space, continuous-time nonlocal dynamics naturally leads to infinite-dimensional integro-differential equations (IDEs), making principled Bayesian inference intractable. We propose the NonLocal Bayesian Spatio-Temporal model (NLBST), a hierarchical Bayesian framework for continuous spatio-temporal fields that learns explicit nonlocal coupling while retaining tractable inference. NLBST represents the latent field via a coordinate-based spatial basis expansion and models the coefficient process with a continuous-time ODE whose learnable linear operator corresponds to a Galerkin reduction of a nonlocal IDE; a Neural ODE residual captures additional nonlinear dynamics. A linear-Gaussian observation model enables Kalman-style sequential updates under missing and irregular observations, while the spatial basis representation enables inductive prediction at unmeasured locations without retraining. Global parameters are learned via variational inference, and uncertainty is handled through a Bayesian hierarchy. Experiments on synthetic and real-world datasets demonstrate strong forecasting and spatial generalization with well-calibrated uncertainty, yielding substantial gains over baselines in strongly nonlocal and partially observed regimes.

The Differential Time-Varying Effect Model (DTVEM) of Jacobson et al. (2019) is a popular tool for finding the best time lag in intensive longitudinal data, but it assumes everyone shares the same lag structure. The original authors named fixing this as future work, and it clashes with the premise of modern clinical research, which is that people differ. We present DTVEM-RE, an extension that lets each person have their own lag coefficients, with two versions of the confirmatory step: a discrete-time hierarchical Bayesian VAR in Stan, which pools across people and gives calibrated uncertainty, and a continuous-time per-person Ornstein-Uhlenbeck model in ctsem, which handles unevenly spaced beeps directly. We report four results. A simulation shows the Bayesian version recovers the between-person spread tau_a with bias below 0.01 and coverage of 90 to 93 percent. On the Fisher et al. (2017) EMA dataset (N=40), person-specific lag-1 effects vary by an order of magnitude across three mood items, the Bayesian and GAMM estimates agree closely (r=0.87 to 0.92), and DTVEM-RE gives the best one-step-ahead prediction among four discrete-time methods. A multi-lag version shows all nine tau_k values have credible intervals excluding zero, and the lag where people differ most changes across items, something lag-1-only methods like mlVAR cannot detect. Finally, the two versions agree almost exactly on person-specific lag-1 estimates (r >= 0.995), differing only as shrinkage predicts. DTVEM-RE is, to our knowledge, the first person-specific implementation of DTVEM-style lag detection, and it contains standard DTVEM as a special case.

We study training-free fixed-length descriptors for multivariate time series and ask not merely whether such a descriptor performs well, but when it can be expected to work at all. Our object of study is $D(τ)$, built from a time-lagged correlation matrix truncated at the Marchenko-Pastur edge so that only signal-bearing eigenvalues survive and classified by cosine similarity to class centroids with zero learned parameters. The central contribution is not the descriptor but a falsifiable applicability criterion for it. Working from a stationary Gaussian VAR(1) model, we argue that $D(τ)$ separates two classes when the signals are approximately stationary and the class information lives in their cross-channel temporal coupling rather than in marginal per-channel power. We derive, semi-formally, three consequences: a distinguishability condition, why the static ($τ=0$) covariance collapses to chance, and why a stationary but power-discriminated paradigm defeats the descriptor. The criterion is operational: a two-part pre-flight test -- an augmented Dickey-Fuller stationarity check and a power-baseline saturation check -- predicts applicability before any training. We validate both halves on a mixed assortment. On four paradigms that satisfy the criterion (Sleep-EDF, BCI-IV-2a, MIT-BIH, ESC-50) the descriptor is competitive with strong baselines at a fraction of their cost, reaching $88.5\pm4.5\%$ under 20-subject leave-one-subject-out on Sleep-EDF on a single CPU thread. On three that violate it -- non-stationary ERPs, and financial-volatility and wearable-stress regimes that are power-discriminated -- it fails exactly as the pre-flight predicts, and these negatives are the more informative half. We are explicit that $D(τ)$ is not the most accurate representation; its value is a compact, training-free embedding whose domain of validity is known in advance.

Temporal systems often exhibit non-stationary behaviour, such as seasonal climate variation or glucose fluctuations in patients with type-1 diabetes. One way to model non-stationarity is through discrete latent regimes, i.e., stationary segments of time. Such systems induce a Markov Switching Model (MSM), a class of Hidden Markov Models with autoregressive dependencies among latent regimes and observed variables. Identifying latent regimes is challenging in the presence of frequent regime switches and nonlinear and non-Gaussian dynamics, particularly when there are instantaneous effects between the variables, e.g., due to slow rates of measurements. In this work, we establish the identifiability of both latent regimes and regime-dependent causal structures under temporal regime dependencies, nonlinear lagged and instantaneous effects, and independent noise from the exponential family. Our identifiability theory subsumes non-temporal mixtures of causal models. Furthermore, we introduce FlowMSM, a regime detection framework that can be paired with any stationary causal discovery method to recover regime-dependent causal structures. Experiments on synthetic benchmarks and a financial economics dataset demonstrate the effectiveness of our approach to detect latent regimes and discover causal structures from non-stationary time series.

Exchange rate forecasting remains a challenging problem, particularly for emerging economies, where the observed time series exhibit pronounced long-memory dependence, nonlinear dynamics, and sensitivity to macro-financial drivers. Classical models such as ARFIMA capture long-range persistence but fail to adequately represent nonlinear relationships, while modern machine learning approaches often neglect the underlying long-memory structure in macroeconomic series. To address this gap, we propose a Neural AutoRegressive Fractionally Integrated Moving Average (NARFIMA) model that integrates ARFIMA-based long-memory modeling with neural networks for nonlinear function approximation, while incorporating exogenous macroeconomic and uncertainty indicators. The framework provides a unified approach for capturing persistence, nonlinear dynamics, and external shocks. We establish asymptotic stationarity of the NARFIMA process and develop conformal prediction intervals for distribution-free uncertainty quantification. Empirical results for BRIC exchange rates show that NARFIMA consistently outperforms a broad range of forecasting benchmarks across multiple horizons, underscoring the importance of explicitly modeling long-memory dependence in exchange rate dynamics. The `narfima' R package provides an implementation of our approach.

Infinite hidden Markov models provide a flexible framework for modeling time-series with structural changes and complex dynamics, without requiring the number of latent states to be specified in advance. This flexibility is achieved through the hierarchical Dirichlet process prior, while efficient Bayesian inference is enabled by the beam sampler, which combines dynamic programming with slice sampling to truncate the infinite state space adaptively. Despite extensive methodological developments, the role of initialization in this framework has received limited attention. This gap is addressed by systematically evaluating initialization strategies commonly used for finite hidden Markov models and assessing their suitability in the infinite setting. Results from both simulated and real datasets show that distance-based clustering initializations consistently outperform model-based and uniform alternatives, the latter being the most widely adopted in the existing literature.

Understanding the temporal dynamics of functional brain connectivity is important for addressing various questions in network neuroscience, such as how connectivity affects cognition and changes with disease. A fundamental challenge is to evaluate whether connectivity truly exhibits dynamics, or simply is static. The most common approach uses sliding-window methods to model functional connectivity over time, and this is often combined with frequentist hypothesis testing frameworks to evaluate dynamics. However, this requires defining appropriate sampling distributions and hyperparameters, such as window length, which imposes specific assumptions on the dynamics. Here, we explore how these assumptions influence the detection of dynamic connectivity, and introduce an alternative approach based on Bayesian hypothesis testing with Wishart processes. This framework estimates uncertainty in the connectivity estimates, and uses this to provide strength of evidence for both dynamic and static connectivity. It encodes assumptions through prior distributions, allowing prior knowledge on the time-dependent structure of connectivity to be incorporated into the model. Using simulations, we compare the two approaches and demonstrate how different assumptions affect the detection of dynamic connectivity. Finally, by applying both approaches to an fMRI working-memory task, we find that conclusions at the group-level increase robustness to modeling choices. Our work highlights the importance of carefully considering modeling assumptions when evaluating dynamic connectivity.

Many complex systems - be they financial, natural, or social - are composed of units - such as stocks, neurons, or agents - whose joint activity can be represented as a multivariate time series. An issue of both practical and theoretical importance concerns the possibility of inferring the presence of a static relationship between any two units solely from their dynamic behaviour. The present contribution aims at tackling such an issue within the framework of traditional hypothesis testing: briefly speaking, our suggestion is that of linking any two units if behaving in a sufficiently similar way. To achieve such a goal, we project a multivariate time series onto a signed graph by i) comparing the empirical properties of the former with those expected under a suitable benchmark and ii) linking any two units with a positive (negative) edge in case the corresponding series shares a significantly large number of concordant (discordant) values. To define our benchmarks, we adopt an information-theoretic approach that is rooted into the constrained maximisation of Shannon entropy, a procedure inducing an ensemble of multivariate time series that preserves some of the empirical properties on average, while randomising everything else. We showcase the possible applications of our method by addressing one of the most timely issues in the domain of neurosciences, i.e. that of determining if brain networks are frustrated or not, and, if so, to what extent. As our results suggest, this is indeed the case, with the major contribution to the underlying negative subgraph coming from the subcortical regions (and, to a lesser extent, from the limbic ones). At the mesoscopic level, the minimisation of the Bayesian Information Criterion, instantiated with the Signed Stochastic Block Model, reveals that brain regions gather into modules aligning with the statistical variant of the Relaxed Balance Theory.

Population-based and distributional optimization methods, from evolution strategies and consensus-based optimization to covariance-matrix adaptation and stochastic gradient methods viewed as distributional dynamics, are widely used for nonconvex or black-box problems, yet their convergence analyses remain fragmented across algorithm-specific techniques. We introduce an operator calculus in which a broad class of such methods, after choosing an appropriate state space and, where necessary, augmenting the state by memory or strategy variables, is described as a composition of three elementary operators (mutation, selection, and recombination) acting on probability measures. Under explicit stability and regularity conditions, the composite operator admits a pre-generator whose continuous-time limit is a transport-reaction-jump (TRJ) PDE that preserves the operator splitting. On this foundation we establish a modular Lyapunov principle. If a state-space Lyapunov function both dissipates under the full generator and controls the relevant search-space gauges, then the state-space Lyapunov functional and the induced search errors decay exponentially. The additive generator structure allows dissipation estimates to be assembled operator by operator, providing a toolkit for certifying convergence of composite mean-field algorithms.

Hidden Markov models are foundational for sequential inference, but their Markovian assumption fails under pathwise constraints such as precedence requirements, visitation cardinalities, or monotonic state progression, which induce long-range dependencies that invalidate standard dynamic programming algorithms. To deal with this, we present Controller-Augmented Hidden Markov Models (CHMMs), a framework that compiles each constraint into a finite-state controller tracking the minimal sufficient history, after which standard forward--backward and Viterbi recursions on the augmented chain compute exact constrained posteriors and maximum a posteriori paths in both discrete and continuous time, the latter through uniformization. We establish four theoretical guarantees: exactness of constrained inference, monotone ascent of constrained EM, inference complexity linear in the controller cardinality, and a total-variation bound under constraint misspecification. A catalog of controller encodings covering 11 constraint families across the ordering, visitation, path, and temporal categories operationalizes the framework. Empirically, we evaluate CHMMs against 6 alternative decoders on 3 real-world sequence-labeling tasks of substantively different character: gene-structure decoding in \emph{Drosophila melanogaster}, free-living activity recognition in CASAS smart-home environments, and protocol-defined human activity recognition from wearable sensors. The results reveal a clean local-versus-cumulative dichotomy in which controller augmentation is uniquely able to recover globally feasible trajectories on cumulative-constraint regimes, whilst simpler decoders are matched in validity on locally-dominated regimes. Together, theory and experiment characterize when exact controller augmentation is necessary and when simpler approaches suffice.

This work makes two advances in the study of the (approximate) nonparametric maximum likelihood estimator (NPMLE) for exponential family mixture models. First, we develop a data-compression strategy that reduces the cost of repeated likelihood evaluations in NPMLE computation to polylogarithmic order in the sample size. Second, we show that, for a broad class of approximate NPMLEs, the resulting marginal density estimator attains an almost parametric rate of convergence.

We study the problem of likelihood maximization when the likelihood function is intractable but model simulations are readily available. We propose a sequential, gradient-based optimization method that directly models the Fisher score based on a local score matching technique which uses simulations from a localized region around each parameter iterate. By employing a linear parameterization to the surrogate score model, our technique admits a closed-form, least-squares solution. This approach yields a fast, flexible, and efficient approximation to the Fisher score, effectively smoothing the likelihood objective and mitigating the challenges posed by complex likelihood landscapes. We provide theoretical guarantees for our score estimator, including bounds on the bias introduced by the smoothing. Empirical results on a range of synthetic and real-world problems demonstrate the superior performance of our method compared to existing benchmarks.

We introduce a non-parametric density estimator deemed Radial Voronoi Density Estimator (RVDE). RVDE is grounded in the geometry of Voronoi tessellations and as such benefits from local geometric adaptiveness and broad convergence properties. Due to its radial definition RVDE is continuous and computable in linear time with respect to the dataset size. This amends for the main shortcomings of previously studied VDEs, which are highly discontinuous and computationally expensive. We provide a theoretical study of the modes of RVDE as well as an empirical investigation of its performance on high-dimensional data. Results show that RVDE outperforms other non-parametric density estimators, including recently introduced VDEs.

Online inventory optimization (OIO) is online convex optimization with physical memory: inventory carryover makes the feasible action set depend on the past. A natural principle, used in stochastic inventory learning and recently in OIO under a single linear capacity constraint, is to maintain a hidden target chosen by an online learner and implement its projection onto the currently feasible order-up-to set. We prove that this simple principle is optimal for OIO on arbitrary bounded convex capacity sets. With online gradient descent as the base learner, the method improves the best known regret guarantee for OIO on general convex sets from inverse to inverse-square-root dependence on the common-demand probability, and we prove a matching lower bound. The same principle gives the first polylogarithmic regret guarantee for strongly convex losses and the first dynamic regret guarantee adapting to Euclidean path variation on general convex capacity sets. The analysis introduces a norm alignment principle: the right state variable is the distance from the hidden target to the feasible set, measured in the same norm as the projection. Under norm alignment, this distance evolves pathwise as a scalar queue, with target movement as arrival and common demand as service. This reduction to one-dimensional queue control resolves the state dependence and extends the guarantees to general convex capacity sets, beyond the reach of prior productwise approaches. Experiments on synthetic and real-world inventory data corroborate the theory.

Clustering is widely used for exploratory analysis and scientific discovery, driving insights from market segmentation to biological data analysis, but its outputs can be difficult to interpret, audit, and reproduce as modern datasets become increasingly large and complex. Reliable use of clustering requires understanding which features drive the discovered structure, yet feature-level explanations for clustering remain scarce compared with methods in supervised learning. Furthermore, existing clustering feature importance scores are often tied to specific algorithms and data assumptions. To address these challenges, we propose Cluster LOCO (Leave-One-Covariate-Out), a family of model-agnostic feature importance scores for clustering. Cluster LOCO is built on feature occlusion and clustering generalizability, defined as whether cluster labels learned on one subset of the data can be accurately predicted on held-out samples. For any chosen clustering algorithm, Cluster LOCO quantifies a feature's importance by measuring how much its removal degrades generalizability. We first introduce Cluster LOCO-Split, which relies on data splitting, and then extend it to Cluster LOCO-MP, a minipatch ensemble-based version designed for large-scale data. Across synthetic simulations and an application to cell-type discovery in single-cell transcriptomics, we show that Cluster LOCO more reliably recovers informative features than existing clustering feature importance methods.

Non-Euclidean optimisation methods with matrix-valued updates, such as Muon and Scion, have recently shown strong empirical performance for training Transformer models, yet their theoretical advantages over Euclidean methods remain poorly understood. We address this gap in the heavy-tailed non-convex regime, where stochastic gradients have bounded $p$-th central moments, $p \in (1,2]$. We show that certain non-Euclidean methods achieve optimal sample complexity under stronger stationarity measures, while Euclidean methods incur additional dimension-dependent costs. As a consequence, for $m \times n$ matrices, Muon finds an $\varepsilon$-stationary point in nuclear norm within $\mathcal{O}\left(\min\{m, n\} \frac{Δ_1 L}{\varepsilon^2} \left(\frac σ\varepsilon \right)^{\frac p {p-1}}\right)$ samples, absorbing heavy-tailed noise without extra dimension dependence, unlike Euclidean methods. We further prove this sample complexity, including its dimension dependence, is optimal for all first-order methods under nuclear-norm stationarity. Experiments on large language models support our theory. Surprisingly, our results suggest that other Schatten geometries beyond the spectral geometry of Muon can perform competitively in certain settings.

Federated learning (FL) often struggles with generalization due to heterogeneous client data. Local models are prone to overfitting their local data distributions, and even transferable features can be distorted during aggregation. To address these challenges, we propose FedCONST, an approach that adaptively modulates update magnitudes based on the parameter strength of the global model. This prevents over-emphasizing well-learned parameters while reinforcing underdeveloped ones. Specifically, FedCONST employs linear convex constraints to ensure training stability and preserve locally learned generalization capabilities during aggregation. A Gradient Signal to Noise Ratio (GSNR) analysis further validates the effectiveness of FedCONST in enhancing feature transferability and robustness. As a result, FedCONST effectively aligns local and global objectives, mitigating overfitting and promoting stronger generalization across diverse FL environments, achieving state-of-the-art performance.

Memorization in diffusion models is often treated as a global property of the model or dataset. In practice, however, a single diffusion model can simultaneously generate both memorized and novel samples. Which training samples are most likely to be memorized? In this work, we show that memorization is governed by \emph{local data coverage}. Leveraging the connection between diffusion models and kernel density estimation (KDE), we derive a theoretical criterion that predicts whether a point is memorized based on the density of training data in its neighborhood and the size of the training dataset. In the high-dimensional limit, this leads to a sharp, local transition: regions of low coverage are dominated by isolated training samples, which are memorized, while dense regions support interpolation and generalization. We validate these predictions empirically, showing that memorization increases with local sparsity and that diffusion models exhibit a coexistence of memorized and novel samples within the same model. Extending this framework to multi-class settings, we further show that classes with higher intra-class sparsity (and thus lower local coverage) are more strongly memorized. Our results provide a local view of memorization in diffusion models, explaining when and where memorization occurs in terms of data geometry.

We develop a statistical learning theory for gradient boosting applied to the estimation of covariate-dependent Generalized Pareto (GP) distributions in the context of Peaks-over-Threshold modeling. After an orthogonal reparametrization of the GP likelihood that diagonalizes its Fisher information matrix, we cast the estimation problem within the Empirical Risk Minimization (ERM) framework and derive non-asymptotic error bounds for the boosting estimator. Our analysis accounts for three distinct sources of error in the process: statistical fluctuations, the approximation bias inherent to the asymptotic nature of the GP model-controlled under second-order regular variation-and the approximation error associated with the finite number of boosting iterates, making explicit the resulting bias-variance trade-off. We illustrate the practical benefits of the reparametrization through simulations, showing that it significantly reduces gradient correlation during training and improves convergence stability. The methodology is applied to a medical malpractice insurance dataset from the Texas Department of Insurance, comprising over 18 000 closed claims. The gradient boosting approach yields a good fit for the tail of settlement cost distributions and reveals that the number of days to settlement is the dominant predictor of tail heaviness, consistent with earlier findings in the reserving literature.

Quantifying the influence of hybrid aleatory and epistemic uncertainties on high-dimensional system responses remains a major challenge in global sensitivity analysis (GSA). Existing Hilbert--Schmidt Independence Criterion (HSIC)-based approaches are primarily restricted to single-output settings and lack a rigorous decomposition of heterogeneous uncertainty sources and their interactions. To address this limitation, a novel double-space tensor-product RKHS framework is proposed for sensitivity analysis under hybrid uncertainty. By constructing factorized kernels over both the latent input space and the multidimensional output space, a concurrent double Möbius inversion is derived to orthogonally decompose the global dependence measure into pure aleatory effects, pure epistemic effects, and their interaction contributions. The resulting dimension-wise sensitivity indices preserve the uncertainty attribution structure across all output dimensions. To satisfy the independence assumptions required by the decomposition, an auxiliary-variable representation based on the inverse probability integral transform is introduced, enabling the treatment of hierarchical uncertainties and Copula-induced correlations within a unified latent space. A fully vectorized single-loop implementation is further developed to avoid the computational burden of nested Monte Carlo simulation. Statistical significance and estimation uncertainty are quantified through permutation testing and Bootstrap confidence intervals. Numerical studies on a modified multi-output Ishigami function and an aerodynamic pressure-field problem demonstrate the accuracy, scalability, and practical applicability of the proposed framework.

In small-batch scientific deployments, labeled target outcomes may be too scarce for reliable shift estimation even when unlabeled target inputs are available. We address the complementary setting where the practitioner has a pre-specified label-shift correction from domain knowledge and asks whether incoming labeled outcomes support it. We show that the per-observation likelihood ratio between a label-shift-corrected predictive and the source predictive is a conditional e-value, so its running product is a nonnegative martingale and Ville's inequality yields an anytime-valid confirmation rule. The log martingale equals the cumulative negative log-predictive density (NLPD) gap between the source and the corrected predictive, converting routine model monitoring into a formal sequential test. Rejection means the incoming data support the posited correction relative to the source predictive, but it is not a precise estimate of the degree of shift. Closed forms are available for GP sources with Gaussian label-shift ratios. GP regression simulations validate Type I control, finite-sample power, miscalibration sensitivity, and the small-batch advantage of a reliable prior over label-based re-estimation.

Optimal Transport has become recently a powerful method for domain adaptation by aligning source and target distributions. We study a supervised domain adaptation problem where source and target domains are related by a rotation or a translation or a homothety in $\mathbb{R}^2$. We prove that the optimal transport map recovers the underlying map when using a $p-$norm cost with $p \geq 2$. Based on this insight, we develop a method combining $K-$means and optimal transport to estimate the underlying map, enabling adaptation of linear regression models when target data is scarce. Simulations demonstrate improved performance over baseline methods. Rather than relying on highly expressive deep learning architectures, we focus on classical machine learning models to emphasize interpretability and theoretical insight. This perspective allows us to explicitly characterize the role of optimal transport in recovering geometric transformations such as rotations, translations, and homotheties. Our contributions include a theoretical result linking optimal transport and rotations, translations and homothecies in $\mathbb{R}^2$, and a practical method for adaptation in linear regression offering both conceptual clarity and applied value in domain adaptation tasks in this space.

Decision trees are one of the fundamental tools in statistical learning due to their interpretability, flexibility, and their ability to adapt to nonlinear structures. Among them, the Classification and Regression Trees, introduced by Breiman, Friedman, Olshen, and Stone in 1984, became one of the most influential algorithms and remains one of the most widely used methods for classification and regression problems. On the other hand, Bregman divergences, introduced by Lev Bregman in 1967 in the context of convex optimization, provide a broad family of loss functions that naturally generalize the squared Euclidean distance. This family includes, among others, the Kullback-Leibler divergence, the Poisson divergence, and the Itakura-Saito divergence, as well as several losses associated with distributions belonging to the exponential family. Moreover, Bregman divergences possess a rich geometric structure and deep connections with convex analysis and information geometry. In this work, we propose a generalization of the CART paradigm based on Bregman divergences, thereby obtaining a broader family of decision trees adapted to different statistical models and underlying geometries. Although algorithms such as CART or classical implementations such as rpart incorporate different impurity criteria, these are usually introduced in an ad hoc manner for each specific model. In contrast, the Bregman divergence approach provides a unified framework that allows these criteria to be derived and interpreted from common convex and geometric principles. Beyond the algorithmic construction, we also investigate theoretical properties of these trees. In particular, we study how properties of the generating convex function -- such as strong convexity or smoothness -- influence impurity gains between parent and child nodes, as well as stability and consistency properties of the estimator.

Sampling plays an important role in long-form language-model reasoning. Over thousands of decoding steps, small changes in the candidate token set can compound into different reasoning trajectories, stability profiles, and final answers. Existing truncation methods such as top-$p$, min-$p$, and fixed top-$nσ$ sampling improve over unrestricted sampling, but they rely on fixed thresholds that cannot adapt to changes in entropy, task difficulty, training stage, or generation budget. We introduce Adaptive Nucleus Truncation Sampling (ANTS), which extends top-\(nσ\) sampling from a fixed decoding rule into an adaptive rollout-control mechanism for long-form generation. ANTS selects standardized neighborhoods around the maximum logit before temperature scaling, adapts the truncation width using an entropy-conditioned controller, and retains a no-truncation fallback arm to stabilize training when truncation becomes unsafe. On a 33B-total / 4B-active sparse Mixture-of-Experts reasoning model, ANTS improves average performance over percentage-based benchmarks by +1.9, +3.8, and +5.2 points at 8K, 16K, and 32K generation budgets, respectively. The strongest gains appear on instruction following and mathematical reasoning, with IFBench improving by more than 10 points at 32K and AIME 2025 improving by 7 points. Code generation reveals an important budget interaction. On Codeforces, ANTS trails the baseline at 8K, but reverses this gap and substantially improves ELO at 16K and 32K. These results suggest that sampler design should be treated not just as a decoding hyperparameter, but as part of how we stabilize and scale long-budget reasoning.

Recursive training of generative models on their own outputs can lead to model collapse, a compounding drift away from the true data distribution. Existing theoretical works bound finite-round error accumulation in the context of diffusion models, but two questions remain open:~what distribution does the recursion converge to, and how fast? We answer both, isolating a mechanism distinct from imperfect learning: even with perfect score estimation and exact sampling, the early stopping of the reverse diffusion (required for numerical stability) drives a progressive drift away from the data distribution. We prove that this recursion converges geometrically to a unique limiting distribution, which admits a closed-form characterization as an infinite mixture of increasingly Gaussian-smoothed versions of the data distribution. A Hermite spectral decomposition of this limit reveals that recursive training acts as a low-pass filter: higher-order modes, which encode fine non-Gaussian structure, are attenuated much more strongly than coarse modes. This spectral picture motivates annealed truncation schedules that progressively shrink truncation times across retraining rounds; we prove that any schedule converging to $0$ asymptotically eliminates recursive compounding. Finally, we show our idealized characterization is robust: in the presence of discretization and score estimation errors, the learned distribution remains in a Wasserstein-2 ball around the ideal limit, with mode-dependent contraction rates that contract high-order errors faster than low-order ones. We validate the theory on synthetic Gaussian mixtures and CIFAR-10.

We study controlled post-training refusal suppression in routed MoE and hybrid-MoE foundation models, aiming to increase non-refusal target-response behavior while preserving general capability under a compact intervention footprint. Existing broad direction-based edits can perturb general-purpose computation, whereas support-only expert edits often lack sufficient capacity to correct heterogeneous refusal representations. To address this limitation, we introduce Localized Multidirectional Correction (LoMC), a support-gated intervention framework that follows a support-then-correction execution order: it first identifies a compact edit support, then aggregates prototype correction directions into layer-wise correction directions, and finally applies rank-one layer-wise correction only within the selected support. By using the edit support as a structural gating constraint, LoMC increases correction capacity without expanding the intervention scope. Experiments on text-only and multimodal safety benchmarks across four routed backbones show that LoMC substantially improves non-refusal target-response behavior while maintaining general capability under a compact intervention footprint.

LLMs utilizing chain-of-thought reasoning often waste substantial compute by producing long, incorrect responses. Abstention can mitigate this by withholding outputs unlikely to be correct. While most abstention methods decide to withhold outputs before or after generation, dynamic mid-generation abstention considers early termination of unpromising reasoning traces at each token position. Prior work has explored empirical variants of this idea, but principled guidance for the abstention rule remains lacking. We present a formal analysis of dynamic abstention for LLMs, modeling abstention as an explicit action within a regularized reinforcement learning framework. An abstention reward parameter controls the trade-off between compute and information. We show that abstaining when the value function falls below this reward strictly outperforms natural baselines under general conditions. We further derive a principled and efficient method to approximate the value function. Empirical results on mathematical reasoning and toxicity avoidance tasks support our theory and demonstrate improved selective accuracy over existing methods.

Activation functions play a central role in neural networks by shaping internal representations. Recently, learning binary activation representations has attracted significant attention due to their advantages in computational and memory efficiency, as well as interpretability. However, training neural networks with Heaviside activations remains challenging, as their non-differentiability obstructs standard gradient-based optimization. In this paper, we propose Heavy Tailed Activation Function (HTAF), a smooth approximation to the Heaviside function that enables stable training with gradient-based optimization. We construct HTAF as a sigmoid hyperbolic tangent composite function and theoretically show that it maintains a large gradient mass around zero inputs while exhibiting slower gradient decay in the tail regions. We show that Spiking Neural Networks, Binary Neural Networks and Deep Heaviside neural Networks can be trained stably using HTAF with gradient-based optimization. Finally, we introduce Implicit Concept Bottleneck Models (ICBMs), an interpretable image model that leverages HTAF to induce discrete feature representations. Extensive experiments across various architectures and image datasets demonstrate that ICBM enables stable discretization while achieving prediction performance comparable to or better than standard models.

End-to-end backpropagation requires storing activations throughout all layers, creating memory bottlenecks that limit model scalability. Existing block-wise training methods offer means to alleviate this problem, but they rely on ad-hoc local objectives and remain largely unexplored beyond classification tasks. We propose $\textit{DiffusionBlocks}$, a principled framework for transforming transformer-based networks into genuinely independent trainable blocks that maintain competitive performance with end-to-end training. Our key insight leverages the fact that residual connections naturally correspond to updates in a dynamical system. With minimal modifications to this system, we can convert the updates to those of a denoising process, where each block can be learned independently by leveraging the score matching objective. This independence enables training with gradients for only one block at a time, thereby reducing memory requirements in proportion to the number of blocks. Our experiments on a range of transformer architectures (vision, diffusion, autoregressive, recurrent-depth, and masked diffusion) demonstrate that DiffusionBlocks training matches the performance of end-to-end training while enabling scalable block-wise training on practical tasks beyond small-scale classification. DiffusionBlocks provides a theoretically grounded approach that successfully scales to modern generative tasks across diverse architectures. Code is available at https://github.com/SakanaAI/DiffusionBlocks .

Although upper bound guarantees for bilevel optimization have been widely studied, progress on lower bounds has been limited due to the complexity of the bilevel structure. In this work, we focus on the smooth nonconvex-strongly-convex setting and develop new hard instances that yield nontrivial lower bounds under deterministic and stochastic first-order oracle models. In the deterministic case, we prove that any first-order zero-respecting algorithm requires at least $Ω(κ^{3/2}ε^{-2})$ oracle calls to find an $ε$-accurate stationary point, improving the optimal lower bounds known for single-level nonconvex optimization and for nonconvex-strongly-convex min-max problems. In the stochastic case, we show that at least $Ω(κ^{5/2}ε^{-4})$ stochastic oracle calls are necessary, again strengthening the best known bounds in related settings. Our results expose substantial gaps between current upper and lower bounds for bilevel optimization and suggest that even simplified regimes, such as those with quadratic lower-level objectives, warrant further investigation toward understanding the optimal complexity of bilevel optimization under standard first-order oracles.

We theoretically investigate the phenomena of generalization and memorization in diffusion models. Empirical studies suggest that these phenomena are influenced by model complexity and the size of the training dataset. In our experiments, we further observe that the number of noise samples per data sample ($m$) used during Denoising Score Matching (DSM) plays a significant and non-trivial role. We capture these behaviors and shed insights into their mechanisms by deriving asymptotically precise expressions for test and train errors of DSM under a simple theoretical setting. The score function is parameterized by random features neural networks, with the target distribution being $d$-dimensional Gaussian. We operate in a regime where the dimension $d$, number of data samples $n$, and number of features $p$ tend to infinity while keeping the ratios $ψ_n=\frac{n}{d}$ and $ψ_p=\frac{p}{d}$ fixed. By characterizing the test and train errors, we identify regimes of generalization and memorization as a function of $ψ_n,ψ_p$, and $m$. Our theoretical findings are consistent with the empirical observations.

Score-based diffusion models are a highly effective method for generating samples from a distribution of images. We consider scenarios where the training data comes from a noisy version of the target distribution, and present an efficiently implementable modification of the inference procedure to generate noiseless samples. Our approach is motivated by the manifold hypothesis, according to which meaningful data is concentrated around some low-dimensional manifold of a high-dimensional ambient space. The central idea is that noise manifests as low magnitude variation in off-manifold directions in contrast to the relevant variation of the desired distribution which is mostly confined to on-manifold directions. We introduce the notion of an extended score and show that, in a simplified setting, it can be used to reduce small variations to zero, while leaving large variations mostly unchanged. We describe how its approximation can be computed efficiently from an approximation to the standard score and demonstrate its efficacy on toy problems, synthetic data, and real data.

In this paper we revisit the classical method of partitioning classification and prove novel convergence rates under relaxed conditions, both for observable (non-privatised) and for privatised data. We consider the problem of classification in a $d$ dimensional Euclidean space. Previous results on the partitioning classifier worked with the strong density assumption (SDA), which is restrictive, as we demonstrate through simple examples. Here, we study the problem under much milder assumptions. We presuppose that the distribution of the inputs is a mixture of an absolutely continuous and a discrete distribution, such that the absolutely continuous component is concentrated on a $d_a$ dimensional subspace. In addition to the standard Lipschitz and margin conditions, a novel characteristic of the absolutely continuous component is introduced, by which the convergence rate of the classification error probability is computed, both for the binary and for the multi-class cases. This bound can reach the minimax optimal convergence rate achievable using SDA, but under much milder distributional assumptions. Interestingly, this convergence rate depends only on the intrinsic dimension of the continuous inputs, $d_a$, and not on $d$. Under privacy constraints, the data cannot be directly observed, and the constructed classifiers are functions of the randomised outcome of a suitable local differential privacy mechanism. In this paper we add Laplace distributed noises to the discretisations of all possible locations of the feature vector and to its label. Again, tight upper bounds on the convergence rate of the classification error probability can be derived, without using SDA, such that this rate depends on $2d_a$.

Off-policy evaluation (OPE) methods estimate the value of a new reinforcement learning (RL) policy prior to deployment. Recent advances have shown that leveraging auxiliary datasets, such as those synthesized by generative models, can improve the accuracy of OPE methods. Unfortunately, such auxiliary datasets may also be biased, and existing methods for using data augmentation within OPE lack principled uncertainty quantification. In high stakes domains like healthcare, reliable uncertainty estimates are important for ensuring safe and informed deployment of RL policies. In this work, we propose two methods to construct valid confidence intervals for OPE with data augmentation. The first provides a confidence interval over $V^π(s)$, the policy value conditioned on an initial state $s$. To do so we introduce a new conformal prediction method suitable for Markov Decision Processes (MDPs) with continuous state spaces, extending prior work to higher-dimensional settings. Second, we consider the more common task of estimating the average policy performance over all initial states, $V^π$; we introduce a method that draws on ideas from doubly robust estimation and prediction powered inference. Across simulators spanning inventory management, robotics, healthcare, and a real healthcare dataset from MIMIC-IV, we find that our methods can effectively leverage auxiliary data and consistently produce confidence intervals that cover the ground truth policy values, unlike previously proposed methods. Our work enables a future in which OPE can provide rigorous uncertainty estimates for high-stakes domains.

Inference-time scaling techniques have significantly bolstered the reasoning capabilities of large language models (LLMs) by harnessing additional computational effort at inference without retraining. Similarly, Chain-of-Thought (CoT) prompting and its extension, Long CoT, improve accuracy by generating rich intermediate reasoning trajectories, but these approaches incur substantial token costs that impede their deployment in latency-sensitive settings. In this work, we first show that truncated CoT, which stops reasoning before completion and directly generates the final answer, often matches the full CoT sampling while using dramatically fewer tokens. Building on this insight, we introduce Fractured Sampling, a unified inference-time strategy that interpolates between full CoT and solution-only sampling along three orthogonal axes: (1) the number of reasoning trajectories, (2) the number of final solutions per trajectory, and (3) the depth at which reasoning traces are truncated. Through extensive experiments on five diverse reasoning benchmarks and several model scales, we demonstrate that Fractured Sampling consistently achieves superior accuracy-cost trade-offs, yielding steep log-linear scaling gains in Pass@k versus token budget. Our analysis reveals how to allocate computation across these dimensions to maximize performance, paving the way for more efficient and scalable LLM reasoning. Code is available at https://github.com/BaohaoLiao/frac-cot.

The need for uncertainty quantification in anomaly detection systems has become increasingly important. In this context, effectively controlling Type I error rates without inflating Type II error rates in these systems can build trust and reduce costs associated with false discoveries. The field of conformal anomaly detection emerges as a promising approach for providing respective statistical and finite-sample validity guarantees through model calibration. However, reliance on calibration data imposes practical limitations, especially in low-data regimes. In this work, we formally define and evaluate leave-one-out-, bootstrap-, and cross-conformal methods for conformal anomaly detection, building on methods from the field of conformal prediction. Looking beyond the classical split-conformal approach, we show that derived methods for calculating resampling-conformal $p$-values offer a practical compromise between the data efficiency of full-conformal (transductive) approaches and the computational efficiency of split-conformal (inductive) methods. We validate derived methods and quantify their improvements for a range of one-class classifiers and datasets.

Background: Understanding whether patients follow similar or distinct patterns of care is important for characterizing clinical practice, identifying patient subgroups, and supporting quality improvement. However, routine healthcare trajectories are difficult to compare directly because patients may differ in their diagnostic workup, treatment sequencing, timing of clinical events, and documentation practices. Despite this variation, trajectories often contain recurring patterns at the cohort level. Methods: To address this challenge, we present a framework that explicitly separates cohort-level typical pathway identification from patient-level inference. At the cohort level, we derive an interpretable representation of care processes using a rule-based algorithm to identify typical healthcare pathways, resulting in a compact pathway graph. These pathways are then modeled as Markov chains and used as structured components in an admixture model, allowing each patient to be represented as a probabilistic mixture of typical pathways rather than being assigned to a single pathway component. The resulting admixture weights provide a compact representation of patient trajectories for subgroup characterization. We further assess the stability of the identified pathways and inferred admixture representations across multiple train-test splits. Results: Across train-test splits, the framework demonstrated consistent pathway structures and patient-level mixture patterns. Applied to routine care data from prostate cancer patients undergoing radical prostatectomy, the framework identified interpretable care patterns and supported the identification of patient subgroups with similar clinical event patterns. Conclusion: Overall, the proposed framework provides an interpretable and stable approach for summarizing treatment pathways and characterizing patient subgroups in real-world practice.

Effective modeling of endangered marine mammal species, such as the North Atlantic Right Whale, is critical for balancing marine conservation with the growing blue economy. Passive acoustic monitoring data collected by autonomous underwater vehicles provide new opportunities for localized marine species detection and oceanographic sensing, but introduce complex statistical challenges such as zero inflation, imperfect detection, and intricate dependence structures. In response, we propose the Deep Zero-Inflated Bernoulli (DeepZIB) model--a deep statistical method which jointly models latent species presence and conditional detection probabilities while learning complex habitat relationships from heterogeneous covariate information. We establish theoretical results on the model's structural properties and conduct simulation experiments to demonstrate its ability to recover underlying parameters and latent presence fields. Application to real-world passive acoustic monitoring data on the North Atlantic Right Whale along the U.S. East Coast demonstrates improved model adequacy and predictive performance in capturing the species' dynamic and spatially varying habitat. A key advantage of DeepZIB is its ability to generate high-resolution, spatially and temporally varying presence maps, providing valuable insights for targeted and risk-aware management of blue economy industries, ranging from offshore and marine energy, to fisheries management and maritime transport.

Several key metrics in public health convey the probability that a primary event will lead to a more serious secondary event in the future.Several key metrics in public health convey the probability that a primary event will lead to a more serious secondary event in the future. These "severity rates" can change over the course of an epidemic in response to shifting conditions like new therapeutics, variants, or public health interventions. In practice, time-varying parameters such as the case-fatality rate are typically estimated from aggregate count data. Prior work has demonstrated that commonly-used ratio-based estimators can be highly biased, motivating the development of new methods. In this paper, we develop an adaptive deconvolution approach based on approximating a Poisson-binomial model for secondary events, and we regularize the maximum likelihood solution in this model with a trend filtering penalty to produce smooth but locally adaptive estimates of severity rates over time. This enables us to compute severity rates both retrospectively and in real time. Experiments based on COVID-19 death and hospitalization data show that our deconvolution estimator is generally more accurate than the standard ratio-based methods, and displays reasonable robustness to model misspecification.

Large language models (LLMs) are increasingly evaluated for bias using adaptations of human psychological paradigms, yet methodological limitations-particularly the conflation of refusal behavior with task performance-have hindered clear interpretation. Here, we adapt the Implicit Association Test (IAT) to a controlled, forced-choice framework and introduce a two-stage modeling approach that separates response compliance from task-consistent classification. Across three contemporary LLMs (Claude Sonnet-4, Gemini 2.5 Pro, and GPT-5), we evaluate associative interference, defined as reduced task-consistency in incongruent relative to congruent conditions. While compliance with the structured response format was uniformly high, interference effects varied substantially across models and domains. Claude Sonnet-4 exhibited strong interference in the Gender--Career domain (DeltaP = 0.086, 95% CrI [0.026, 0.173]) and smaller but credible effects in Gender--Science. Gemini 2.5 Pro showed attenuated interference, and GPT-5 exhibited minimal or no detectable interference across domains. These findings demonstrate that IAT-style associative asymmetries are not a universal property of LLMs, but instead depend on model-specific characteristics. By isolating interference from compliance and modeling item-level variability, this study provides a principled framework for evaluating structured response patterns in LLMs. The results highlight the importance of model-specific assessment and suggest that associative interference can be substantially mitigated in modern systems.

Political scientists often interpret coefficient shrinkage under fixed effects as evidence that pooled associations are confounded. This paper shows why that inference is unreliable for slow-moving, mismeasured regressors. Fixed effects can remove much of the signal and identify coefficients from within-unit variation that is disproportionately measurement error, attenuating estimates toward zero. A lone fixed effects coefficient may therefore be unable to distinguish confounding from measurement error. I show that the attenuation depends on a regressor's empirical intraclass correlation and measurement reliability. I then propose a default workflow for panel regression. Researchers estimate reliability when possible, report pooled and fixed effects estimates with corrected within reliability, use partial identification bounds when the estimates share a sign, and report fixed effects as a within-unit estimate when they do not. For variables with no reliability estimate, I introduce an autocorrelation frontier that bounds the attenuation factor directly. I conclude by applying this workflow to several published results to show that the data often cannot distinguish attenuation from confounding, and the workflow makes clear which case the researcher faces.

The multiperiod mean-variance (MV) portfolio optimization serves as a vital expansion of Markowitz's static MV portfolio selection framework. Just like its static counterpart, the multiperiod MV portfolio remains susceptible to estimation errors. We propose a reference-regulated multiperiod mean-variance (RRMV) framework that penalizes deviations from a reference policy. Therefore, this new optimization successfully combines the advantages of dynamic strategies and reference portfolios. A key contribution of this paper is the characterization of the out-of-sample Sharpe ratio under high-dimensional asymptotics with estimation errors in both the mean vector and the covariance matrix. We show how the reference penalty and the investment horizon jointly affect the optimized portfolio performance, and how regularization operates differently from the single-period portfolio optimization. Extensive simulation and real data studies demonstrate that the proposed framework improves the stability and out-of-sample Sharpe ratios of multiperiod policies significantly.

Understanding how a prediction model will perform in a new environment before deployment is essential to preventing harm when algorithms inform decision-making. Two common sources of model performance degradation are (i) covariate shift, where the target covariate distribution differs from the source, and (ii) selective labels, where the observability of outcomes depends on historical decisions. We study pre-deployment model evaluation under the joint presence of covariate shift and labeling of outcomes selectively based on observed features. In particular, we present a double machine learning procedure for estimating the target risk of an arbitrary black-box prediction model under a general loss function. We show identification of this estimand under standard assumptions and derive a bias-corrected estimator based on the influence function of the target risk. Finally, we evaluate our estimator through experiments using the eICU electronic health records database, showing that it tracks the true target risk more accurately than methods that address either selective labels or covariate shift alone, as well as baselines that combine standard plug-in approaches.

Synthetic data has become a prominent solution for preserving privacy while sharing data, but current empirical risk assessment frameworks fundamentally assume a sample-based context that fails to translate for the evaluation of synthetic population level datasets. This commentary explores the implications when synthesizing entire populations in order to do population-level data science, arguing that traditional metrics, such as Membership Inference Attacks (MIA) and Attribute Inference Attacks (AIA), require re-examination. First, MIA may be rendered irrelevant in contexts where population membership is public knowledge or not considered sensitive information. Second, the risk of singling out is heightened because the confidential data contain full population information. Additionally, the absence of an "out-of-sample" comparison group for attribute inference means we need to define other policies when defining acceptable inferences. Finally, we cannot rely on simply returning to subsampling prior to generating synthetic data if the use case is truly to enable population-level data science. This commentary highlights the necessity for considering context when generating and evaluating synthetic population data.

We propose a sequential test for detecting arbitrary distribution shifts that allows conformal test martingales (CTMs) to work under a fixed, reference-conditional setting. Existing CTM detectors construct test martingales by continually growing a reference set with each incoming sample, using it to assess how atypical the new sample is relative to past observations. While this design yields anytime-valid type-I error control, it suffers from test-time contamination: after a change, post-shift observations enter the reference set and dilute the evidence for distribution shift, increasing detection delay and reducing power. In contrast, our method avoids contamination by design by comparing each new sample to a fixed null reference dataset. Our main technical contribution is a robust martingale construction that remains valid conditional on the null reference data, achieved by explicitly accounting for the estimation error in the reference distribution induced by the finite reference set. This yields anytime-valid type-I error control together with guarantees of asymptotic power one and bounded expected detection delay. Empirically, our method detects shifts faster than standard CTMs, providing a powerful and reliable distribution-shift detector.

This study presents a statistical and machine learning framework for characterizing a hydrogen-based multi-energy system (H-MES) using one year of high-resolution operational data. Statistical analysis revealed a binary operation driven by renewable surplus, with solar irradiance explaining 45.7% of rank-based variance in hydrogen production, a large effect by conventional standards. Only high-irradiance periods triggered meaningful electrolyzer engagement, while electricity demand exerted a weaker inverse suppression effect ($ε^2 = 0.126$). Multiple regression confirmed electrolyzer power as the dominant linear predictor, with a synergistic solar-wind interaction. Notably, Random Forest analysis ranked wind output first in predictive importance despite its weak bivariate correlation (r = 0.167), revealing non-linear dynamics invisible to parametric methods. A sequence model exploited strong 24-hour autocorrelation (r = 0.845) for operational forecasting, while a reinforcement learning agent optimized hydrogen revenue dispatch. The core contribution is demonstrating that statistical and machine learning approaches are complementary for H-MES modeling and control.

The R package bpvars was designed to forecast employment, unemployment, and labour market participation rates of 189 countries. However, it is generally applicable to dynamic panel data due to the flexibility of its modelling framework and robust coding. It includes a family of Bayesian hierarchical panel Vector Autoregressions (VARs) that are characterised by: (i) country-specific VAR models (ii) with their parameters' priors centred around their global counterparts, and (iii) featuring flexible multi-level hierarchical prior distributions (iv) with many variants of well-established in the literature benchmark choices, and (v) four alternative specifications including groupping of country-specific or global parameters. A~distinguishing feature is its implementation of missing observation treatment based on a model-coherent Bayesian approach. These models are accompanied by Bayesian prediction, offering a wide range of possible specifications that aim to increase forecasting precision and comply with various reporting standards. We also implement pseudo-out-of-sample recursive forecasting for evaluating point and density forecast performance. The package implements model specification, estimation, and forecasting routines, facilitating simple workflows and reproducibility, including estimation and forecasting results summaries and visualisations. It achieves extraordinary computational speed thanks to the employment of frontier econometric and numerical techniques, as well as algorithms written in C++.

Coarse-grained (CG) molecular simulations offer an efficient alternative to atomistic molecular dynamics to study large and complex biological systems. The accuracy of CG simulations has been increased dramatically by the introduction of machine-learned coarse-grained (MLCG) models. However, these models are typically designed to be used at a single thermodynamic point, lack temperature transferability, and can not be used to predict temperature dependent quantities like the heat capacity. Here we introduce a thermodynamically informed, temperature-transferable MLCG framework for proteins that explicitly decomposes the CG potential of mean force (PMF) into its energetic and entropic components. The model architecture enforces an exact thermodynamic relation between the energetic and entropic components of the PMF and guarantees physically consistent extrapolation and interpolation across temperature regimes. We validate this framework on an extensive dataset spanning a total of 250 $μ$s of molecular dynamics simulations across five temperatures between 300 K and 400 K for the Chignolin protein, and demonstrate that it reproduces the temperature dependency of the reference atomistic free energy surfaces, correcting the temperature-unaware baselines. Furthermore, we show that it is possible to apply an inexpensive, post-hoc temperature-dependent correction that does not require retraining the MLCG potential, accurately recovering the atomistic heat capacity at different temperatures. Overall, this work provides a physically grounded pathway toward thermodynamically transferable MLCG simulations of complex biomolecular systems.

The ability to resolve complex physical phenomena with high fidelity and at low computational cost is central to addressing key challenges in modern engineering. A prime example lies in hypersonic flows, where the precise prediction of the full flowfield topology, in particular with respect to shock wave location and intensity, is critical. Yet supersonic and hypersonic flows continue to be a stumbling block for traditional reduced-order models and neural emulators that struggle to capture steep gradients in flow states with physical consistency in applications of industrial relevance. To that end, we introduce a fully GPU based workflow that integrates accelerated data generation with the training of neural emulators augmented by uncertainty quantification and physics-aware refinement. Our workflow is enabled by a differentiable high-fidelity solver (JAX-Fluids) which we employ for rapid dataset creation and residual-based improvement of the neural emulator to enhance physical consistency. Building on this framework, we first present a suite of model architectures and analyze their scaling behavior to expose their strengths and shortcomings. We then show that residual-based refinement enables training on cases where only mesh and input parameters are available, substantially reducing residuals and improving physical consistency. Together, differentiable simulation and residual-based refinement yield physics emulators that remain reliable beyond their training distribution, a key requirement for deploying surrogates in real-world engineering design loops.

We introduce a Process Oriented Guided Inquiry Learning (POGIL)-style activity for teaching Bayesian reasoning in introductory statistics through conditional probability, Bayes' theorem, and belief updating. The activity is self-contained, uses hand-computable probabilities organized in two-way tables, and engages students in structured team roles. We evaluated the activity in four sections of an undergraduate introductory statistics course using a quasi-experimental comparison of POGIL-style and lecture-based instruction for a Bayes' theorem unit. Outcomes included student performance on Bayes' theorem final exam questions and satisfaction with instruction. We used a Bayesian bivariate generalized linear model to compare the two approaches while accounting for major type, gender, and race. The results indicated similar exam performance and similar probabilities of high satisfaction across instructional styles and demographic groups, with considerable uncertainty and no clear evidence of meaningful differences. These findings suggest that the POGIL-style activity performed comparably to lecture-based instruction for this unit while offering an active and classroom-ready way to introduce Bayesian reasoning without requiring difficult computation or simulation. We provide adaptable instructional materials and a reproducible Bayesian analytic framework for evaluating active learning innovations in introductory statistics. Our study supports the feasible inclusion of Bayesian reasoning in introductory courses and may help instructors considering active learning.

The Legacy Survey of Space and Time (LSST) on the Vera C. Rubin Observatory will generate a massive collection of time series (light curves) of the measured flux of transient and variable astronomical objects. With each new flux observation, light curve classifiers need to generate updated probability distributions over candidate classes, which will then be shared with the global community for the purpose of identifying interesting targets for follow-up observations as well as less time-sensitive analysis applications. Using the synthetic light curves and classification results of participating classifiers from the Extended LSST Astronomical Time-series Classification Challenge (ELAsTiCC), we investigate a novel framework to enhance existing light curve classifications by incorporating their classification histories and the temporal evolution of these histories. To demonstrate the potential of this approach, we introduce a model that combines a recurrent neural network and an additive attention module, which shows improved classification accuracy and more balanced precision-recall performance compared to existing classifiers from the challenge. Furthermore, at this stage, most, if not all, of the existing classifiers are evaluated by their final classification results on complete light curves; we propose new metrics that evaluate the stability, accuracy, and early classification performance of a classifier's predictions when using limited data by considering the Wasserstein distance between the temporally evolving classification probability distributions. Our metrics offer a more comprehensive perspective for model assessment by supplementing classical methods such as the confusion matrix and precision-recall.

Better representation of the uncertainty in a data visualisation is a focus of recent research activity. A problem with the current literature is that there is a lack of clarity about the definition of uncertainty and what it means to represent it in a plot. This confusion results in a significant amount of conflicting results in the literature, especially in experiments that assess the effectiveness of different uncertainty representations. In this review, we summarise the current literature, provide workable definitions, and illustrate these definitions with examples. In doing so, we ask what it really takes to achieve transparency in statistical graphics. It is hoped that it will be useful for guiding new graphics methodology and experimental research.

Accurate aerodynamic prediction often relies on high-fidelity simulations; however, their prohibitive computational costs severely limit their applicability in data-driven modeling. This limitation motivates the development of multi-fidelity strategies that leverage inexpensive low-fidelity information without compromising accuracy. Addressing this challenge, this work presents a multi-fidelity deep learning framework that combines autoencoder-based transfer learning with a newly developed Multi-Split Conformal Prediction (MSCP) strategy to achieve uncertainty-aware aerodynamic data fusion under extreme data scarcity. The methodology leverages abundant Low-Fidelity (LF) data to learn a compact latent physics representation, which acts as a frozen knowledge base for a decoder that is subsequently fine-tuned using scarce HF samples. Tested on surface-pressure distributions for NACA airfoils (2D) and a transonic wing (3D) databases, the model successfully corrects LF deviations and achieves high-accuracy pressure predictions using minimal HF training data. Furthermore, the MSCP framework produces robust, actionable uncertainty bands with pointwise coverage exceeding 95%. By combining extreme data efficiency with uncertainty quantification, this work offers a scalable and reliable solution for aerodynamic regression in data-scarce environments.

Lehner (1999) derived elegant formulas for the operator norm $\|\mathfrak{X}\|$ of operators of the form $\mathfrak{X} = \mathbf{A}_0 \otimes \mathfrak{1} + \sum_{i = 1}^n \mathbf{A}_i \otimes \mathfrak{m}_i$, also easily generalized to the spectral edge $λ_{\max}(\mathfrak{X})$, in terms of nonlinear optimization problems over positive definite matrices. Here the $\mathbf{A}_i$ are finite-dimensional Hermitian matrices, the $\mathfrak{m}_i$ are either free semicircular or free Rademacher families of operators, and $\mathfrak{1}$ is the identity operator. We first show that both of Lehner's nonlinear optimizations can be rewritten as linear semidefinite programs (SDPs), even in the Rademacher case where Lehner's optimization is not itself convex. We give the primal and dual forms of these SDPs, derive the complementary slackness relations and consequences thereof, and propose that the SDPs are more stable and accurate than the iterative numerical scheme proposed in Lehner's original work. We then apply the SDPs from the semicircular case to spiked matrix models, studied recently via Lehner's formula by Bandeira, Cipolloni, Schröder, and van Handel (2024). We give a new proof of the Baik--Ben Arous--Péché (BBP) transition they establish in models with isotropic (but possibly correlated) Gaussian noise by constructing feasible variables for the associated primal and dual SDPs. Combining our construction with a sensitivity interpretation of optimal dual variables, we study the fluctuations of leading eigenvectors of such models. We conjecture and give numerical evidence that these fluctuations are Gaussian but anisotropic and non-universal, and that their covariance may be computed in terms of the optimizer of the dual of Lehner's formula, which in turn is approximately the leading eigenmatrix of a completely positive operator associated to the covariance of the noise model.

We propose a kernel-based estimator of the velocity field governing the transport and diffusion of $d$-dimensional interacting particles. Assuming the initial positions are i.i.d. with law $μ_0$, we establish consistency of the estimator with an explicit mean-squared error rate of order $h^2 + N^(-2/(d+2))$, where $h$ denotes the time-discretization step and N the number of particles. The analysis covers two distinct settings: the Fokker-Planck equation, where we recover the underlying potential function, and the McKean-Vlasov equation, where we deconvolve the interaction kernel driving the mean-field dynamics.

Let \((w_{ij})_{i,j\ge1}\) be a single infinite array of independent identically distributed real- or complex-valued entries of mean zero, variance \(σ^2\), and finite fourth moment. Set \(W_n=(w_{ij})_{1\le i,j\le n}\) and \(X_n=n^{-1/2}W_n\). For every fixed \(k\ge1\), we identify the almost sure limiting operator norm of several fixed products built from this family. Define the \(k\)-th freeness coefficient by \[ γ_k:=\sqrt{\frac{(k+1)^{k+1}}{k^k}}. \] Then we prove \[ \|X_n^k\|\toσ^kγ_k \qquad \text{almost surely}. \] The same limit holds for products sampled with replacement from any fixed finite pool of independent copies of \(X_n\); in particular, it holds for the product of \(k\) independent copies. Thus, the freeness coefficient captures the non-commuting characteristic between large random matrices %powers and independent or fixed-pool sampled products under the finite fourth moment assumption. The improvement of the classical Bai--Yin-type power estimate from the scale \(σ^k(k{+}1)\) to \(σ^k \sqrt{k{+}1}\) is a direct corollary of our result. The main technical challenge is to prove the upper bound using a high-moment expansion of %the upper bound is proved by a high-moment expansion of \(\E\Tr((X_n^kX_n^{*k})^m)\). The leading zero-defect trace words are tree-like and are counted by the Fuss--Catalan number \[ F_{k,m}= \frac1{km+1}\binom{(k+1)m}{m}. \] The combinatorial tool helps to devise a defect-sensitive global enumeration: if \(L=km\) and \[ r=(L+1-v)+(L-q), \] then the number of admissible word classes with defect \(r\) is at most \(F_{k,m}(Cm)^{Dr}\). This polynomial-in-\(m\) loss, with degree proportional to the defect, is summable in the logarithmic moment range.

Recovering structural information from noisy high-dimensional data is a fundamental task in statistical inference. We investigate the recovery thresholds for a graph hidden in a randomly weighted complete graph. Specifically, an unknown graph $H^* \in H_n$ is chosen uniformly at random, and hidden in a complete graph of $n$ vertices as follows: the weight of an edge $e \in H$ is distributed independently according to $P_n$; otherwise the weight is distributed independently according to $Q_n$. The goal is to recover almost all of $H$ from these edge weights. Assuming a local Lipschitzness of the Rényi divergence between distributions $P_n$ and $Q_n$, and a mild density condition for the graphs $H_n$, we give a unified characterization of the information-theoretic limit for recovering almost all of $H$ (also known as almost exact recovery). Our characterization connects the KL divergence between $P_n$ and $Q_n$ to the logarithm of the first moment threshold of $H$ in the Erdős-Rényi random graph model $G(n,p)$. Our lower bound also extends to the task of partial recovery, in which only a constant $λ$-fraction of $H$ needs to be recovered. Last but not least, for certain Bernoulli and Exponential regimes, and for Gaussian distributions, we are able to show an All-or-Nothing (AoN) threshold phenomenon at the exponential scale.

We consider the problem of estimating the gradient ascent line of a density originating at a given point. Going beyond mere consistency, we establish a weak convergence result for a plugin estimator based on a kernel density estimator of the density. We then leverage that result to construct a confidence region for the gradient ascent line, including by bootstrap.

The analysis of non stationary signals in complex physical systems often relies on Time Frequency distributions. Among these, the Pseudo Wigner Ville Distribution (PWVD) stands out for its superior resolution but is mathematically challenging due to its inherent quadratic nonlinearity. This nonlinearity generates complex interference artifacts and cross terms in the phase space, potentially obscuring the physical features of the signal, particularly for nonlinear chirps. In this work, we establish a mathematically grounded framework for the PWVD for general windowed nonlinear chirps. By leveraging the theory of oscillatory integrals with coalescing stationary points, we derive a uniform asymptotic expansion that bridges the gap between heuristic signal processing and semiclassical geometric approaches (Berry's chord construction). The resulting closed form representation, expressed in terms of symmetric incomplete Airy functions, provides a unified description of the nonlinear transform's behavior, regularizing the transition across the instantaneous frequency caustics. While the framework is general, we show its power on two illustrative examples: the high precision nonlinear chirps of coalescing binaries in gravitational-wave astronomy and radar nonlinear chirps for pulse compression applications. The analytical results successfully predict the structure of interference patterns and quantify the systematic bias in peak based frequency estimation. Therefore, this study establishes a systematic bridge between nonlinear mathematical analysis and precision experimental physics, validating the PWVD as a robust tool for detailed source characterization in high signal to noise regimes.

Redox-flow battery (RFB) research spans molecular design, electrolyte optimization, electrode and membrane materials, stack operation, system management, and safety analysis, making it a constrained, multi-scale, and multi-objective energy-storage R&D problem. Although large language models (LLMs) can support scientific knowledge integration and proposal generation, generic LLM reasoning remains insufficiently adaptive across innovation-oriented exploration, rule-based execution, mechanistic modeling, and system-level trade-offs. Here we introduce ChargeBD, a character-aware heterogeneous-agent reasoning framework for guided engineering in battery development. Starting from a 50-question RFB-specific task set, we construct a 500-question ESS-LLM Benchmark and define MBTI-inspired persona agents as structured cognitive-bias templates rather than psychometric instruments or representations of real personalities. DeepSeek-V3-Plus is selected as the shared base model, and 16 MBTI-inspired persona agents are evaluated to construct a persona capability matrix and a cognitive advantage matrix.

This paper investigates asymptotic minimaxity properties of Bayesian multiple testing rules in the sparse Gaussian sequence model using a broad class of global-local scale mixtures of normals as priors for the means. Minimaxity is studied under standard misclassification loss and the composite loss given by the sum of the false discovery proportion (FDP) and false non-discovery proportion (FNP). When the sparsity level is known, we show that by suitably choosing the global shrinkage parameter based on the sparsity level, our proposed testing rule achieves the exact minimax risk asymptotically for both losses under the ''beta-min'' separation condition. When the sparsity level is unknown, both empirical Bayes and fully Bayesian adaptations of the same rule are shown to achieve exact minimax risk asymptotically under suitable assumptions on sparsity. Our results reveal that minimaxity is attained for ''horseshoe-type'' priors that are broad enough to include the horseshoe, Strawderman-Berger, standard double Pareto, and certain inverse-gamma priors, among others. For non-''horseshoe-type'' priors, minimaxity fails to hold for either loss function. To the best of our knowledge, these are the first results of their kind for multiple hypothesis testing based on global-local shrinkage priors.

Dense Associative Memory (DAM) models generalize the classical Hopfield model by incorporating n-body or exponential interactions that greatly enhance storage capacity. While the criticality of DAM models has been largely investigated, mainly within a statistical equilibrium picture, little attention has been devoted to the temporal self-organizing behavior induced by learning. In this work, we investigate the behavior of a stochastic exponential DAM (SEDAM) model through the lens of Temporal Complexity (TC), a framework that characterizes complex systems by intermittent transition events between order and disorder and by scale-free temporal statistics. Transition events associated with birth-death of neural avalanche structures are exploited for the TC analyses and compared with analogous transition events based on coincidence structures. We systematically explore how TC indicators depend on control parameters, i.e., noise intensity and memory load. Our results reveal that the SEDAM model exhibits regimes of complex intermittency characterized by nontrivial temporal correlations and scale-free behavior, indicating the spontaneous emergence of self-organizing dynamics. Notably, such regimes arise over finite intervals of noise intensity rather than at a single critical point, consistent with the concept of extended criticality. Further, the noise intensity range needed to reach the critical region, where self-organizing behavior emerges, slightly decreases as the memory load increases. This study highlights the relevance of TC as a complementary framework for understanding learning and information processing in artificial and biological neural systems, revealing the link between the memory load and the self-organizing capacity of the network.

This paper introduces a marketing decision framework that optimizes customer targeting by integrating heterogeneous treatment effect estimation with explicit business guardrails. The objective is to maximize revenue and retention while adhering to constraints such as budget, revenue protection, and customer experience. The framework first estimates Conditional Average Treatment Effects (CATE) using uplift learners, then solves a constrained allocation problem to decide whom to target and which offer to deploy. It supports decisions in retention messaging, event rewards, and spend-threshold assignment. Validated through offline simulations and online A/B tests, the approach consistently outperforms propensity and static baselines, offering a reusable playbook for causal targeting at scale.

We introduce two families of stochastic interventions with discrete treatments that connect causal modeling to cost-sensitive decision making. The interventions arise from a cost-penalized information projection of the independent product of the organic propensity scores and a reference policy, yielding closed-form Boltzmann-Gibbs couplings. The induced marginals define modified stochastic policies that interpolate smoothly, via a tilt parameter, from the organic law or from the reference law toward a product-of-experts limit when all destination costs are strictly positive. The first family recovers and extends incremental propensity score interventions, retaining identification without global positivity. For inference on the expected outcomes after these policies, we derive the efficient influence functions under a nonparametric model and construct one-step estimators. In simulations, the proposed estimators improve stability and robustness to nuisance misspecification relative to plug-in baselines. The framework can operationalize graded scientific hypotheses under realistic constraints. Because inputs are modular, analysts can sweep feasible policy spaces, prototype candidates, and align interventions with budgets and logistics before committing experimental resources.

Time series anomaly detection presents various challenges due to the sequential and dynamic nature of time-dependent data. Traditional unsupervised methods frequently encounter difficulties in generalization, often overfitting to known normal patterns observed during training and struggling to adapt to unseen normality. In response to this limitation, self-supervised techniques for time series have garnered attention as a potential solution to undertake this obstacle and enhance the performance of anomaly detectors. This paper presents a comprehensive review of the recent methods that make use of self-supervised learning for time series anomaly detection. A taxonomy is proposed to categorize these methods based on their primary characteristics, facilitating a clear understanding of their diversity within this field. The information contained in this survey, along with additional details that will be periodically updated, is available on the following GitHub repository: https://github.com/Aitorzan3/Awesome-Self-Supervised-Time-Series-Anomaly-Detection.

The recovery of causal effects in structural models with missing data often relies on $m$-graphs, which assume that missingness mechanisms do not directly influence substantive variables. Yet, in many real-world settings, missing data can alter decision-making processes, as the absence of key information may affect downstream actions and states. To overcome this limitation, we introduce $lm$-SCMs and $lm$-graphs, which extend $m$-graphs by integrating a label set that represents relevant context-specific independencies (CSI), accounting for mechanism shifts induced by missingness. We define two causal effects within these systems: the Full Average Treatment Effect (FATE), which reflects the effect in a hypothetical scenario had no data been missing, and the Natural Average Treatment Effect (NATE), which captures the effect under the unaltered CSIs in the system. We propose recovery criteria for these queries and present doubly-robust estimators for a graphical model inspired by a real-world application. Simulations highlight key differences between these estimands and estimation methods. Findings from the application case suggest a small effect of ADHD treatment upon test achievement among Norwegian children, with a slight effect shift due to missing pre-tests scores.

In cognition, response times and choices in decision-making tasks are commonly modeled using Drift Diffusion Models (DDMs), which describe the accumulation of evidence for a decision as a stochastic process, specifically a Brownian motion, with the drift rate reflecting the strength of the evidence. In the same vein, the Poisson counter model describes the accumulation of evidence as discrete events whose counts over time are modeled as Poisson processes, and has a spiking neurons interpretation as these processes are used to model neuronal activities. However, these models lack a learning mechanism and are limited to tasks where participants have prior knowledge of the categories. To bridge the gap between cognitive and biological models, we propose a biologically plausible Spiking Neural Network (SNN) model for decision-making that incorporates a learning mechanism and whose neurons activities are modeled by a multivariate Hawkes process. First, we show a coupling result between the DDM and the Poisson counter model, establishing that these two models provide similar categorizations and reaction times and that the DDM can be approximated by spiking Poisson neurons. To go further, we show that a particular DDM with correlated noise can be derived from a Hawkes network of spiking neurons governed by a local learning rule. In addition, we designed an online categorization task to evaluate the model predictions. This work provides a significant step toward integrating biologically relevant neural mechanisms into cognitive models, fostering a deeper understanding of the relationship between neural activity and behavior.

Confounding bias and selection bias bring two significant challenges to the validity of conclusions drawn from applied causal inference. The latter can stem from informative missingness, such as in cases of attrition. We introduce the Sequential Adjustment Criteria (SAC), which extend available graphical conditions for recovering causal effects from confounding and attrition using sequential regressions, allowing for the inclusion of post-exposure and forbidden variables in the adjustment sets. We propose an estimator for the recovered Average Treatment Effect (ATE) based on Targeted Minimum-Loss Estimation (TMLE), which exhibits multiple robustness under certain technical conditions. This approach ensures consistency even in scenarios where the Double Inverse Probability Weighting (DIPW) and the naive plug-in sequential regressions approaches fall short. Through a simulation study, we assess the performance of the proposed estimator against alternative methods across different graph setups and model specification scenarios. As a motivating application, we examine the effect of pharmacological treatment for Attention-Deficit/Hyperactivity Disorder (ADHD) upon the scores obtained by diagnosed Norwegian schoolchildren in national tests using observational data ($n=9\,352$). Our findings align with the accumulated clinical evidence, affirming a positive but small impact of medication on academic achievement.

We define the Probability of Tiered Benefit in scenarios with a binary exposure and an outcome that is either categorical with $K \geq 2$ ordered tiers or continuous partitioned by $K-1$ fixed thresholds into disjoint intervals. Similarly to other pure counterfactual queries, this parameter is not $g$-identifiable without additional assumptions. We demonstrate that strong monotonicity does not suffice for point identification when $K \geq 3$ and provide sharp bounds both with and without such constraint. Inference and uncertainty quantification for these bounds are challenging tasks due to potential nonregularity induced by ambiguities in the underlying individualized optimization problems. Such ambiguities can arise from immunities or null treatment effects in subpopulations with positive probability, affecting the lower bound estimate and hindering conservative inference. To address these issues, we extend the available Stabilized One-Step Correction (S1S) procedure by incorporating stratum-specific stabilizing matrices. Through simulations, we illustrate the benefits of this approach over existing alternatives. We apply our method to estimate bounds on the probabilities of tiered benefit and harm from pharmacological treatment for ADHD upon academic achievement, employing observational data from diagnosed Norwegian schoolchildren. Our findings indicate that while girls and children with low prior test performance could have moderate chances of both benefit and harm from treatment, a clear-cut recommendation remains uncertain across all strata.