Papers

It is well known in random matrix literature that the limiting spectral distribution of a Wigner matrix is the semi circular law while the limiting spectral distributions of other patterned matrices like Toeplitz, Hankel, symmetric circulant and reverse circulant matrices have unbounded supports. One fundamental difference between the Wigner matrices and the other matrices mentioned above is that the Wigner matrices have $O(n^2)$ independent random variables while the others have $O(n)$ independent random variables. In this paper, we show that this is not true in general. In particular, we form matrices with $O(n)$ independent random variables whose empirical spectral distributions are arbitrary close to the semi circular law.

One-dimensional Gaussian processes with stationary, integrable kernel functions admit exact or arbitrarily accurate state-space representations, enabling linear-time inference through Kalman filtering and Rauch-Tung-Striebel (RTS) smoothing. However, the RTS smoother requires inversion of predicted state covariance matrices, which can become ill-conditioned and may therefore lead to numerical instabilities. In this work, we revisit the modified Bryson-Frazier (MBF) smoother as an alternative to the RTS smoother for Gaussian process regression in its state-space representation. In addition to reducing computational cost and memory requirements, the MBF smoother computes the same posterior distributions as the RTS smoother while avoiding the problematic covariance matrix inversion and the associated numerical instabilities. Furthermore, we demonstrate that the intermediate quantities computed by the MBF smoother can be reused to compute gradients of the negative log marginal likelihood, enabling kernel hyperparameter learning with minimal additional cost. Together, these results establish the MBF smoother as a unified and numerically robust approach to inference and kernel hyperparameter learning for one-dimensional Gaussian process regression.

In high-dimensional online prediction, sparse comparators motivate regret bounds that depend on sparsity rather than ambient dimension. Feature priming seeks such adaptation by reweighting features using past data and refitting a minimum-norm predictor. At COLT 2023, Warmuth and Amid posed the open problem of whether the univariate, Pearson, or multivariate priming rules admit competitive online regret guarantees. Under the natural past-only Moore–Penrose protocol, we establish sparse-regret lower bounds that refute the corresponding sparse-logarithmic guarantee. The key obstruction is cheap nuisance interpolation, which permits exact interpolation of the history while assigning insufficient weight to the truly predictive coordinate. An exact target-mass identity and a two-sign argument convert this obstruction into clipped prediction loss. Hadamard constructions yield $\Omega(\min\{T,\sqrt d\})$ clipped regret for each of the three unit-power rules against a zero-loss one-sparse comparator. For every fixed power $\alpha\ge1$, one shared paired construction further yields linear regret simultaneously for all three powered rules and selectors among them in sufficiently high dimension. A rank upper bound is tight for powered univariate priming, even with Euclidean-unit inputs, and for unit-power Pearson priming with coordinatewise bounded inputs and target-preserving totalization. A separate algebraic construction gives $\Omega(\min\{T,d^{1/4}\})$ regret for unit-power multivariate priming under Euclidean-unit inputs. The univariate lower bound persists under any nonnegative second-stage ridge schedule, while a paired ridge construction yields linear lower bounds for all three powered rules. Exploratory diagnostics on frozen language-model activations are consistent with the same qualitative mechanism. The exact multivariate frontier remains open.

Inverse learning within a statistical framework has a wide range of applications. It has garnered significant attention in machine learning, artificial intelligence, and related fields, where the goal is to infer unknown parameters from indirect and noisy observations. This work investigates the stable approximation of $u^{\dagger}$ which solves the equation $Au=g$, with $A$ being a linear operator between appropriate vector spaces. We will consider the domain to be a non-reflexive Banach Space and the co-domain to be a space of real-valued functions on a metric space $X$. The function $g$ is characterized by a finite number of independently and identically distributed data points, which are assumed to follow some unknown probability measure $\rho$. We employ Tikhonov regularization with an arbitrary convex functional to obtain the regularized solution corresponding to the given data point. The convergence analysis is carried out with respect to the Bregman distance, and an upper bound for the error is derived in probability terms. The theoretical findings are then supported by numerical experiments.

We study uniform sampling of binary matrices with fixed row and column sums, a recurring problem in ecological null models, Rasch-model testing, network analysis, and combinatorics. We propose the Snake algorithm, a rejection-free Markov chain Monte Carlo sampler that grows an alternating path until its first self-intersection and flips the resulting loop. The chain is reversible and irreducible on the fixed-margin state space, hence has the uniform stationary distribution. We prove that one step flips on the order of $\sqrt{n}$ entries in sparse and balanced square regimes, give upper bounds on the per-step path length, and show that the resulting work per flipped entry is rate optimal in sparse and balanced regimes and near-optimal up to a polylogarithmic factor under a one-sided half-balanced condition. A Markov-chain comparison, combined with the recently established universal spectral-gap bound for the swap chain, proves that the lazy Snake chain is rapidly mixing for every feasible pair of margins; in the permutation-matrix case, the raw chain has the sharp total-variation mixing time $\Theta(n \log n)$. We also describe a directed-graph extension and an equal-margin label-shuffling variant. Numerical experiments against Swap, Rectangle Loop, Curveball, sequential importance sampling, and a directed edge-swap algorithm show consistent gains in move size, wall-clock convergence, and sampling efficiency.

Regularized sparse regression has been extensively studied in the offline setting, but online formulation remains relatively under-explored. This gap stems from four key challenges: (i) the infeasibility of dynamically updating the regularization parameter in every online round, (ii) managing storage and memory complexity, (iii) enabling real-time computation via closed-form updates rather than solving full optimization problems at each round, and (iv) achieving optimal statistical guarantees under realistic assumptions. In this paper, we propose an online generalized-sparsity-constrained regression framework, focusing on online cardinality-constrained linear regression and low-rank matrix sensing. Unlike online regularized regression, our constrained formulation eliminates the need for dynamic parameter tuning. We introduce an efficient online hard-thresholding algorithm that performs closed-form updates and requires storing only summary statistics, making it computationally, memory, and storage efficient. Despite the inherent nonconvexity and combinatorial nature of the formulation, our algorithm achieves global convergence at the optimal statistical rate under realistic assumptions, provided that the projection set is properly overparameterized. Numerical experiments demonstrate that our method consistently outperforms state-of-the-art alternatives.

Learning restricted Boltzmann machines (RBMs) is computationally challenging because it requires expectations whose exact evaluation is generally intractable. The expectations are typically evaluated using a sampling approximation based on blocked Gibbs sampling (BGS), which is a local Markov chain Monte Carlo transition kernel. However, the locality of BGS can lead to poor sampling quality when the RBM has high energy barriers, thereby degrading learning performance. Deep tempering (DT), which performs parallel tempering over a sequence of learnable RBMs including the training RBM, alleviates this locality issue. However, DT algorithmically requires multiple steps to move through the RBM sequence to achieve a nonlocal transition. In this paper, we propose a transition kernel defined over the RBM sequence used in DT. The proposed kernel has a round-trip structure over the sequence, enabling nonlocal moves within a single transition while leaving the RBM sequence invariant. Numerical experiments show that the proposed kernel performs nonlocal transitions more frequently and achieves higher sampling quality with fewer transitions than BGS and DT. We further verify that learning based on the proposed kernel is more stable and mitigates the training failures observed with BGS- and DT-based learning.

Successive Difference Replication (SDR) is a replication based method of variance estimation introduced by Fay and Train (1995) for estimators based on complex multistage surveys, especially those including a final systematic sampling stage. The method has been used for many years as the primary variance-estimation methodology in large national household surveys administered by the Census Bureau, including the American Community Survey and also the Current Population Survey's monthly estimates based on self-representing strata. In settings where it is applied, generally no second method of variance estimation has been available, so the performance of SDR has been studied via simulation by various authors, for variances of survey totals and of nonlinear survey estimators. This paper begins with a thorough exposition of the SDR method and review of previously published results on the small-domain biases of SDR variance estimation. It is shown that the number D of cycles used in implementing SDR should be 3 or larger, in order to control the variability of SDR estimates, but need not be larger than 5. Beyond that, the value of D is virtually irrelevant to the occurrence of small-domain bias in SDR. The SDR method is shown via theoretical formulas and simulation to inflate average estimated variances in small domains by amounts that vary systematically with the patterns of attribute means and variances and survey weights in consecutively enumerated strata. The degree of average variance inflation is generally moderate, no more than 15 percent in domains with sample size 20, but can be larger in special settings. Moreover, SDR estimates are extremely variable in small domains, with standard deviations often far larger than any biases.

Data-driven algorithm design frames hyperparameter tuning as a statistical learning problem, but establishing generalization guarantees remains challenging due to the implicit, non-smooth dependence of model performance on hyperparameters. Existing multi-dimensional bounds under piecewise-polynomial assumptions remain theoretically loose and lack comprehensive lower bounds. We resolve this by establishing tight pseudo-dimension bounds for multi-dimensional data-driven tuning. First, we refine the learning-theoretic upper bound using real algebraic geometry; by analyzing invariant connected sign cells during block elimination rather than isolated sign vectors, we avoid topological over-counting to derive strictly sharper sample complexities. Second, we present a multi-regime lower-bound framework that disentangles combinatorial and algebraic capacities. By constructing shattered problem instances across distinct regimes, we prove our upper bounds are tightly saturated. Finally, we extend our topological framework to accommodate general bi-level validation-loss tuning and broader semi-algebraic applications.

Modern probabilistic time-series forecasters often express uncertainty through forecast samples. While typically converted into nominal prediction regions using empirical quantiles, these model-implied sets lack formal coverage guarantees and frequently deviate from nominal targets under distribution shift. Existing multivariate conformal methods can calibrate these regions online, but they typically estimate geometry from historical residuals using fixed or accumulating look-back windows. This reliance on the past limits their ability to exploit the instantaneous dependence structure of current predictions and leaves them vulnerable to stale-regime contamination. To address this, we propose SPACE, a conformal wrapper for sample-generating multivariate forecasters. SPACE constructs ellipsoidal joint prediction regions by estimating time-local covariance geometry directly from the current forecast sample cloud, calibrating the region's radius via a dynamic backward window-selection scheme. Across diverse multivariate datasets, probabilistic forecasters, and conformal baselines, SPACE consistently brings realized joint and rolling coverage closer to the nominal target, achieving superior coverage-efficiency tradeoffs relative to competing wrappers.

In large-scale epidemiological studies, it is often of interest to investigate joint relationships between longitudinal and time-to-event outcomes with exposures that are trajectories or functions. Our motivation study is the Objective Physical Activity and Cardiovascular Health (OPACH) Study, which collected accelerometry-measured physical activity in 6,489 older women. One of the scientific aims is to understand sedentary behavior accumulation patterns and its association with physical function (longitudinal) and mortality (time-to-event). We propose a novel approach that first converts raw accelerometry data into daily sitting bout accumulation profiles, which are treated as functional covariates, and then develop a functional joint model for longitudinal and time-to-event outcomes that incorporates a baseline functional covariate. The longitudinal process is modeled using functional data methods and linked to the survival process through functional principal component scores. Both sub-models include interpretable linear scalar-on-function regression coefficients to capture flexible dose-response associations between sitting accumulated across varying bout durations and health outcomes. Estimation is carried out via an efficient expectation-maximization (EM) algorithm with penalized spline approximations. Simulation studies demonstrate accurate parameter estimation and reliable model selection. Application to the OPACH data reveals flexible and interpretable dose-response relationships between sitting bout durations, physical function, and mortality.

We study finite-sample parameter estimation in logistic regression with Gaussian design, where the goal is to estimate $\mathbf{\theta}^*\in \mathbb{R}^d$ with $R=\|\mathbf{\theta}^*\|_2\ge 1$ from i.i.d. samples $\{(\mathbf{x}_i,y_i)\}_{i=1}^n,$ $\mathbf{x}_i \sim N(0,\mathbf{I}_d)$, $y_i\mid \mathbf{x}_i \sim \mathrm{Bernoulli}((1+\exp(-\mathbf{x}_i^\top \mathbf{\theta}^*))^{-1})$. In this paper, we provide the first minimax optimal estimator, and improve on the best known finite-sample error rate for the maximum likelihood estimator (MLE). These two accomplishments are due to a minimax optimal estimator for the parameter norm $R$. First, we establish the minimax lower bound $\Omega(\sqrt{R^3/n})$ for norm estimation. We then improve the best known norm estimation error rate of the MLE, i.e., $O(\sqrt{R^3d/n})$ from Chardon, Lerasle and Mourtada (2024), to $\tilde{O}(\sqrt{R^3/n}+R^2d/n)$. The additional term, $R^2d/n$, appears to be the intrinsic bias of the MLE, as evidenced by the high-dimensional asymptotic theory of Zhao, Sur and Candes (2022) and numerical examples. We show that, however, this additional term is not information-theoretically necessary. To this end, we construct an efficient debiased norm estimator that achieves the error rate $O(\sqrt{R^3/n})$ and is therefore minimax optimal. Combining this with the optimal direction estimator given by the MLE, we establish the minimax optimal rate $\Theta(\sqrt{Rd/n}+\sqrt{R^3/n})$ for estimating $\mathbf{\theta}^*$, as well as the improved finite-sample error rate $\tilde{O}(\sqrt{Rd/n}+\sqrt{R^3/n}+R^2d/n)$ for the MLE. Numerical experiments demonstrate that the proposed minimax optimal estimators outperform the MLE.

This paper develops an adaptive surrogate modeling method for problems with very high-dimensional spatio-temporal outputs. The analysis of spatio-temporal multi-physics systems is computationally expensive and consists of a large number of inputs and outputs. Surrogate models are often constructed to replace the physics-based model to achieve computational efficiency in analyses such as uncertainty quantification and optimization that require many function calls. In order to address the challenge introduced by the high dimensionality of spatio-temporal output, a dimension reduction method is first employed to map the high-dimensional output to a low-dimensional latent space. This is followed by the construction of the surrogate model in the low-dimensional space. The prediction error in the original space, which includes both the reconstruction error and surrogate model error, is evaluated using different error metrics. Based on the prediction accuracy of the surrogate model, new training points are identified for adaptive improvement of the surrogate model. We present a novel adaptive sampling technique that combines exploration and exploitation to improve the surrogate model accuracy with the fewest possible runs of the expensive physics-based model. Thermo-mechanical analysis of a gas turbine engine blade is used to analyze the effectiveness of the proposed method.

When computational models (either physics-based or data-driven) are used for the sensitivity analysis of engineering systems, the sensitivity estimate is affected by the accuracy and uncertainty of the model. This paper considers global sensitivity analysis (GSA) for situations where both a physics-based model and experimental observations are available, and investigates physics-informed machine learning strategies to effectively combine the two sources of information in order to maximize the accuracy of the sensitivity estimate. Two representative machine learning (ML) techniques are considered, namely, deep neural networks (DNN) and Gaussian process (GP) modeling, and two strategies for incorporating physics knowledge within these techniques are investigated, namely: (i) incorporating loss functions in the ML models to enforce physics constraints, and (ii) pre-training and updating the ML model using simulation and experimental data respectively. Four different models are built for each type (DNN and GP), and the uncertainties in these models are included in the Sobol indices computation. The DNN-based models, with many degrees of freedom in terms of model parameters and training options, are found to result in smaller bounds on the sensitivity estimates when compared to the GP-based models. The proposed methods are illustrated for additive manufacturing and lake temperature modeling examples.

This article investigates several physics-informed and hybrid machine learning strategies that incorporate physics knowledge in experimental data-driven deep-learning models for predicting the bond quality and porosity of fused filament fabrication (FFF) parts. Three types of strategies are explored to incorporate physics constraints and multi-physics FFF simulation results into a deep neural network (DNN), thus ensuring consistency with physical laws: (1) incorporate physics constraints within the loss function of the DNN, (2) use physics model outputs as additional inputs to the DNN model, and (3) pre-train a DNN model with physics model input-output and then update it with experimental data. These strategies help to enforce a physically consistent relationship between bond quality and tensile strength, thus making porosity predictions physically meaningful. Eight different combinations of the above strategies are investigated. The results show how the combination of multiple strategies produces accurate machine learning models even with limited experimental data.

This paper addresses the limitations of Gaussian distribution assumptions in statistical sparse learning, particularly in modeling correlated and heterogeneous data. Conventional Gaussian models often lack robustness towards outliers and underlying distribution assumptions. To overcome these limitations, we propose the use of the $q$Gaussian distribution, derived from Tsallis entropy maximization, as a robust alternative. This is notably relevant in biostatistics, where the presence of correlated observations and heterogeneity, such as in genetic and longitudinal studies, is prevalent. Our contributions include modeling of correlated data through the re-derived multivariate probability density function from Tsallis entropy maximization, thereby addressing the limitations inherent in conventional Gaussian models. Furthermore, we introduce a novel framework that adapts numerical methods designed to find equilibria in flows to tackle composite optimization problems prevalent in statistical sparse learning. Applying this framework to the Hager-Zhang conjugate gradient algorithm \cite{Hager2005}, we develop a numerically stable and efficient algorithm for sparse statistical learning. The $q$Gaussian distribution, informed by the principle of maximizing Tsallis entropy, presents a viable and flexible alternative to Gaussian-based methods. This paper not only contributes to the theoretical understanding of statistical distributions and optimization techniques, but also paves the way for practical data analysis.

Financial returns are heavy-tailed, and accurate tail risk estimation is central to portfolio risk management. Modern neural generators sample by pushing a simple base distribution through a learned map, and for training stability that map is built from Lipschitz components. This is the binding constraint: a Lipschitz map of a Gaussian is sub-Gaussian, so heavier-tailed targets admit no exact match at any finite Lipschitz constant. The Monge–Ampère equation ties the Brenier map's local distortion to the density ratio $f/(g\circ T)$, so a deeper trough in the target density requires a higher-gain map and yields a higher-variance estimator. The argument needs only bounded distortion, so it covers normalizing flows, flow matching, GANs, and diffusion samplers alike. Semi-Discrete Optimal Transport (SDOT) relaxes the map's regularity rather than the source's tail class. Its power diagram gives every training observation a cell holding exactly $1/N$ of the source measure, and tail observations are reached by crossing a cell boundary rather than by stretching. Our primary experiment sweeps severity over a calibrated Merton jump-diffusion spanning kurtosis 94 to 1,679. SDOT holds tail ratios at $0.85$–$0.94$ with cross-seed standard deviations below $0.025$, while every learned generator either compresses the tails or inflates them with a variance that grows alongside. Further experiments carry the result to real S&P~500 returns and to a 21-year backtest, where SDOT gives the best risk-adjusted market-neutral strategy under CVaR optimization (Sharpe $0.70$, max drawdown $-2.60\%$, against $0.40$ for the next-best generator).

Modern supply chain networks increasingly rely on real-time data to navigate structural uncertainties, market volatility, and operational disruptions. This manuscript bridges the gap between statistical data-driven learning and robust decision-making frameworks in logistics and operations management. We present a comprehensive, mathematically rigorous treatment of supply chain analytics, moving from empirical demand forecasting to optimal inventory and network control under uncertainty. Key topics explored include sample minimization, dynamic programming recursions for time-varying inventory replenishment, network fulfillment frameworks, and advanced distributionally robust optimization (DRO) via transport theory to hedge against rare events. By integrating predictive statistical models with prescriptive control algorithms, such as column generation for vehicle routing and non-homogeneous queueing regimes, this text provides the foundational tools necessary for designing resilient, data-driven automated systems. It serves as both a theoretical blueprint and an algorithmic guide for researchers and practitioners operating at the intersection of machine learning, mathematical optimization, and applied probability.

Customer decline early-warning systems feed account-manager action lists, and every flagged account consumes intervention capacity. In a deployed business-to-business marketplace system, one action-list slot in three went to flags that dissolve under a seasonally aligned label. The standard target in non-contractual churn prediction compares an entity's next k months of activity with its trailing k months. The two windows cover different calendar months, so for seasonal entities the threshold-ratio construction confounds seasonality with decline, and the event rate depends on the label's anchor calendar month. We formalize the mechanism and measure it on three public panels and the production panel (the public arms pre-specified). Of the adjacent-window decay events, 37–69% on the public panels and 28–50% in production have no counterpart under a seasonally aligned definition. Pooling anchors, the standard remedy, balances the training mixture but corrects no individual label. Measured alternatives repair the curve only partially or change the detection horizon. Aligning the baseline to the same k calendar months one year prior – advice practitioners already state, here formalized, measured, and costed – flattens the curve at the source. With the classifier held fixed, production holdout ROC-AUC rises from 0.767 to 0.864 for decline (different targets; the gap closes on synthetic ground truth and a production hindsight referee). The served action list shrinks by a third, 119 to 79 accounts. The measured price is extra history, a blind spot to decline-then-stabilization, and a stricter cut under sustained growth.

This paper challenges the pessimistic meta-inductive argument against scientific realism by undermining its inductive step rather than its historical premise. Although related challenges already exist, I develop a new one. Drawing on a general epistemology of scientific inference developed in frequentist statistics, machine learning, and formal epistemology, I evaluate induction in terms of convergence to the truth. I argue that ordinary enumerative induction can achieve everywhere convergence, whereas meta-induction fails even to achieve almost everywhere convergence. Indeed, in the problem context where meta-induction arises, the failure is deeper: no inference method whatsoever achieves almost everywhere convergence.

Contrastive learning has become a cornerstone of modern representation learning, powering CLIP-style models that underpin text-to-image generation, vision-language models, and retrieval across a rapidly growing range of modalities. Despite this empirical success, the expressive power of these architectures remains poorly understood. To gain insight, we study expressivity by adopting a population-level, density-estimation viewpoint: each architecture comprises a parameterized set of densities whose parameters may be chosen to approximate the joint distribution of the modalities. This isolates a question of pure representational capacity: which joint distributions can a given contrastive family of parameterizations approximate to arbitrary accuracy? We show that expressivity is sharply architecture-dependent. For two modalities, the simple two-tower CLIP architecture is a universal approximator. A natural generalization of CLIP, widely used in practice when three or more modalities are present, is based on a loss found by summing over all pairwise similarities. This provably cannot represent arbitrary joint distributions, although we prove that it remains expressive enough to match all pairwise conditionals. Motivated by this gap, we propose Hadamard-CLIP, which adds a single learned weight vector on top of the existing encoders and restores universal approximation of the joint for any number of modalities while preserving CLIP's fast, precomputable-embedding retrieval.

This paper proposes a nonparametric estimation procedure for extropy and its extensions, namely the {\alpha}-Rényi and {\alpha}-Tsallis extropies, for finite discrete random variables. We establish almost sure rates of convergence and asymptotic normality for the plug-in estimators. The theoretical results are validated through a comprehensive simulation study. The findings provide a solid foundation for the use of extropy-based measures in practical applications, including forecasting, risk assessment, and decision-making under uncertainty.

Every team building Large Language Models (LLMs) faces a core challenge: offline benchmarks show performance gains and user engagement rises post deployment, but isolating cause from effect remains difficult. Simultaneous marketing, media coverage, and seasonal demand obscure whether model updates truly drive engagement gains. This paper presents a novel causal estimation approach that leverages non uniform quality improvements across capabilities within a single model version. Because capabilities improve unevenly (e.g., strong gains in coding versus modest gains in writing), users experience varied quality depending on their task distribution. This variation in experienced quality provides causal signal for estimation. We validate this approach on synthetic data with known ground truth. The raw estimator recovers 81% to 88% of the true effect, with the remainder lost to measurement error in usage estimates. Applying an errors in variables disattenuation adjustment (using a test retest reliability ratio mean correlation of 0.861) corrects the estimate to 1.017 (bootstrapped 95% CI: 0.915 to 1.109). By contrast, naive methods fail significantly, recovering only 63% without version controls and 62% using real time rather than frozen usage patterns. Permutation tests confirm the framework distinguishes true causal effects from noise. Sensitivity analysis indicates recovery improves monotonically from 78% to 87% as pre period length increases from 3 to 7 weeks, highlighting a clear tradeoff between sample duration and estimator precision. Seed sensitivity tests further confirm stability across random draws.

This paper studies the finite-sample bias of plug-in estimators for Shannon, Rényi, and Tsallis extropies for finitely supported discrete random variables. We derive explicit first-order bias formulas for all three estimators by analyzing the bias of the intermediate functional S_\alpha = \sum_{i=1}^r (1-p_i)^\alpha. Bias-corrected estimators are then proposed. We show that the uncorrected and corrected estimators are asymptotically equivalent, differing only by terms of order O(1/n). A simulation study validates the theoretical results and demonstrates the superior finite-sample performance of the bias-corrected estimators. The results justify the use of the simpler uncorrected plug-in estimators in asymptotic settings, such as those considered in the companion paper on almost sure convergence and asymptotic normality.

Online decision making often requires navigating a landscape shaped by both dynamic contexts and strategic interactions. In competitive pricing, for example, hotels must account for both dynamic contextual factors and rivals' strategic responses. Existing approaches address only part of this challenge: contextual bandits optimize single-agent decisions using observable features but ignore multi-player interactions, while online matrix games capture strategic behavior through Nash equilibrium but assume fixed payoffs, ignoring contextual information. How should agents act then when strategic payoffs evolve with contextual signals? We introduce online contextual matrix games to integrate contextual information into multi-player online games. We further propose OnGameLearn, an online learning algorithm that efficiently balances exploration and exploitation across both player actions and contexts. This approach comes with statistical guarantees: tail bounds for the estimated payoff matrix, the convergence of the estimated Nash equilibrium, the asymptotic normality of the parameter estimators, and the sublinear regret bound. We also develop the notion of policy value in matrix games and develop a doubly robust, $\sqrt{T}$-consistent estimator for it. Across simulated studies and a real-world hotel pricing application, we find that OnGameLearn effectively navigates the intertwined challenges of strategic and contextual decision-making.

Active inference selects actions by minimising an expected free energy functional over predicted futures. However, adding an expectation over yet-unobserved outcomes means the free energy functional no longer has a Kullback-Leibler structure, which hinders message passing treatments of inference procedures. We propose an alternative formulation based on a Bethe free energy functional, fully supporting inference by message passing. The epistemic drive is maintained by imposing an information constraint, next to normalisation, marginalisation and form constraints, insisting that the mutual information between future observations, states and parameters given actions must be at least as large as the entropy of the goal prior. For a specific value of the corresponding Karush-Kuhn-Tucker multiplier, the stationary point of this constrained Bethe Lagrangian recovers the expected free energy solution. We show that, as the information demand is varied, the solved multiplier moves through its inactive, interior, and saturated regimes. In the inactive regime the agent's epistemic drive switches off entirely, while in the saturated regime it is maximal. We compare the performance of the constrained Bethe agent on three tasks against EFE and Q-MDP.

Quasi-Monte Carlo sampling is a numerical integration method that uses points with a space-filling property in $[0,1]^s$ designed to give better estimates than plain Monte Carlo methods do. For integrands of bounded variation in the sense of Hardy and Krause, errors of $O(n^{-1+\epsilon})$ for any $\epsilon>0$ are obtained from $n$ sample points. Randomized quasi-Monte Carlo (RQMC) points are individually uniformly distributed but collectively space-filling and then independent replications provide variance estimates. For smooth enough integrands the randomization can give a root mean squared error of $O(n^{-3/2+\epsilon})$. This article explains RQMC for a statistical readership recounting some history and presenting some current directions.

Recovering the directed acyclic graph (DAG) of a structural equation model (SEM) from observational data is a central problem in causal discovery. The iterative gradient descent and per-problem hyperparameter tuning of continuous-optimization methods are poorly suited to two practically important regimes: the sample-limited regime, where the number of samples is comparable to or smaller than the number of nodes in the DAG, and the compute-limited regime. This work proposes SURE-Ridge, a non-iterative, closed-form estimator for equal variance linear Gaussian SEM. The method performs parallel node-wise regressions with regularization parameters chosen adaptively by Stein's unbiased risk estimate (SURE), and applies an adaptive thresholding procedure to extract a DAG from the resulting soft adjacency matrix. Numerical results show that SURE-Ridge achieves the lowest structural Hamming distance in the small-sample regime and the lowest run time across all sample sizes tested, compared with NOTEARS, DAGMA, and GBNSL baselines.

Electoral reforms – including changes to district boundaries, electoral rules, and the size of elected bodies – are a recurrent feature of American state and municipal politics. The representational impact of such reforms is challenging to assess before they are implemented, and even well-intentioned civic reformers are often surprised by the unintended consequences of changes to the political system. We develop a statistical framework for the prospective evaluation of electoral reforms by integrating several modes of analysis: descriptive statistics of the voter file, probabilistic race and ethnicity imputation, ecological inference, and redistricting simulations via ensembles. Together, these tools provide a unified assessment of how proposed reforms affect whose voices are ultimately heard in government. We apply this framework to reforms currently being considered for the Los Angeles City Council, including increasing the number of single-member districts, adopting ranked-choice voting, and introducing multimember districts elected by proportional ranked choice voting. We find that expanding the Council alone is unlikely to substantially improve representation for underrepresented communities. In contrast, eliminating low-turnout primary elections and adopting multimember districts produce larger improvements in representational equity. Our analyses were shared with L.A.'s Charter Reform Commission via public testimony in late 2025. Although motivated by Los Angeles, the framework is broadly applicable to evaluating prospective electoral reforms across American jurisdictions.

Estimation of wildlife population size and spatial dynamics is central to ecology and conservation. Spatial capture-recapture (SCR) models estimate abundance, using detections at sensors such as camera traps, by linking detection probability to the distance between detectors and latent individual activity centres. However, standard SCR formulations assume detections are conditionally independent over time given activity centres without explicitly modelling movement between detection events. This assumption can be problematic when individuals exhibit movement-driven dependence in detections, potentially leading to biased inference on population size. We address this important issue by developing a continuous-time framework that integrates movement into spatial capture-recapture. Individual movement is modelled as a continuous-time Markov chain over a discretised landscape, and detections arise as state-dependent Poisson events. This yields a Markov-modulated marked Poisson process representation, in which detections provide information about an individual's latent location at the time of observation and allow likelihood-based inference in continuous time. We show through simulation studies that ignoring movement-driven dependence can lead to positively biased estimates of population size, whereas the proposed model recovers unbiased estimates and provides additional inference on space use. An application to camera-trap data of American martens illustrates how the framework yields new insights into movement and density. These results demonstrate that explicitly modelling movement is critical for reliable inference in spatial capture-recapture studies.