Papers

Reinforcement Learning (RL) lacks benchmarks that enable precise, white-box diagnostics of agent behavior. Current environments often entangle complexity factors and lack ground-truth optimality metrics, making it difficult to isolate why algorithms fail. We introduce Synthetic Monitoring Environments (SMEs), an infinite suite of continuous control tasks. SMEs provide fully configurable task characteristics and known optimal policies. As such, SMEs allow for the exact calculation of instantaneous regret. Their rigorous geometric state space bounds allow for systematic within-distribution (WD) and out-of-distribution (OOD) evaluation. We demonstrate the framework's benefit through multidimensional ablations of PPO, TD3, and SAC, revealing how specific environmental properties - such as action or state space size, reward sparsity and complexity of the optimal policy - impact WD and OOD performance. We thereby show that SMEs offer a standardized, transparent testbed for transitioning RL evaluation from empirical benchmarking toward rigorous scientific analysis.

With the rise of high-dimensional correlated data, multicollinearity poses a significant challenge to model stability, often leading to unstable estimation and reduced predictive accuracy. This work proposes the Single-Parametric Principal Component Selection Operator (SPPCSO), an innovative penalized estimation method that integrates single-parametric principal component regression and $L_{1}$ regularization to adaptively adjust the shrinkage factor by incorporating principal component information. This approach achieves a balance between variable selection and coefficient estimation, ensuring model stability and robust estimation even in high-dimensional, high-noise environments. The primary contribution lies in addressing the instability of traditional variable selection methods when applied to high-noise, high-dimensional correlated data. Theoretically, our method exhibits selection consistency and achieves a smaller estimation error bound compared to traditional penalized estimation approaches. Extensive numerical experiments demonstrate that SPPCSO not only delivers stable and reliable estimation in high-noise settings but also accurately distinguishes signal variables from noise variables in group-effect structured data with highly correlated noise variables, effectively eliminating redundant variables and achieving more stable variable selection. Furthermore, SPPCSO successfully identifies disease-associated genes in gene expression data analysis, showcasing strong practical value. The results indicate that SPPCSO serves as an ideal tool for high-dimensional variable selection, offering an efficient and interpretable solution for modeling correlated data.

Understanding the intricate non-convex training dynamics of softmax-based models is crucial for explaining the empirical success of transformers. In this article, we analyze the gradient flow dynamics of the value-softmax model, defined as ${L}(\mathbf{V} \sigma(\mathbf{a}))$, where $\mathbf{V}$ and $\mathbf{a}$ are a learnable value matrix and attention vector, respectively. As the matrix times softmax vector parameterization constitutes the core building block of self-attention, our analysis provides direct insight into transformer's training dynamics. We reveal that gradient flow on this structure inherently drives the optimization toward solutions characterized by low-entropy outputs. We demonstrate the universality of this polarizing effect across various objectives, including logistic and square loss. Furthermore, we discuss the practical implications of these theoretical results, offering a formal mechanism for empirical phenomena such as attention sinks and massive activations.

Differential diagnosis among parkinsonian syndromes remains a clinical challenge due to overlapping motor symptoms and subtle gait abnormalities. Accurate differentiation is crucial for treatment planning and prognosis. While gait analysis is a well established approach for assessing motor impairments, conventional methods often overlook hidden nonlinear and structural features embedded in foot clearance patterns. We evaluated Topological Data Analysis (TDA) as a complementary tool for Parkinsonism classification using foot clearance time series. Persistent homology produced Betti curves, persistence landscapes, and silhouettes, which were used as features for a Random Forest classifier. The dataset comprised 15 controls (CO), 15 idiopathic Parkinson's disease (IPD), and 14 vascular Parkinsonism (VaP). Models were assessed with leave-one-out cross-validation (LOOCV). Betti-curve descriptors consistently yielded the strongest results. For IPD vs VaP, foot clearance variables minimum toe clearance, maximum toe late swing, and maximum heel clearance achieved 83% accuracy and AUC=0.89 under LOOCV in the medicated (On) state. Performance improved in the On state and further when both Off and On states were considered, indicating sensitivity of the topological features to levodopa related gait changes. These findings support integrating TDA with machine learning to improve clinical gait analysis and aid differential diagnosis across parkinsonian disorders.

We study high-dimensional drift estimation for Lévy-driven Ornstein–Uhlenbeck processes based on discrete observations. Assuming sparsity of the drift matrix, we analyze Lasso and Slope estimators constructed from approximate likelihoods and derive sharp nonasymptotic oracle inequalities. Our bounds disentangle the contributions of discretization error and stochastic fluctuations, and establish minimax optimal convergence rates under suitable choices of tuning parameters in a high-frequency regime. We further quantify the sample complexity required to attain these rates depending on the Lévy noise. The results extend the theory of high-dimensional statistics for stochastic processes to a substantially broader class of noise mechanisms, in particular pure jump processes. They also demonstrate that Lasso and Slope remain competitive for jump-driven systems, providing practical guidance for inference in applications where Lévy processes are a natural modeling choice.

Predictive coding graphs (PCGs) are a recently introduced generalization to predictive coding networks, a neuroscience-inspired probabilistic latent variable model. Here, we prove how PCGs define a mathematical superset of feedforward artificial neural networks (multilayer perceptrons). This positions PCNs more strongly within contemporary machine learning (ML), and reinforces earlier proposals to study the use of non-hierarchical neural networks for ML tasks, and more generally the notion of topology in neural networks.

This review provides a systematic overview of methods that combine covariate-based clustering of observational units (patients) with outcome models for clinical studies. We distinguish between informed-cluster models, where the outcome contributes to cluster formation, and agnostic-cluster models, where clustering is performed solely on covariates in a separate first step. Informed-cluster models include product partition models with covariates (PPMx), finite mixtures of regression models (FMR), and cluster-aware supervised learning (CluSL). Agnostic-cluster models encompass two-step procedures using either model-based or algorithmic clustering followed by cluster-specific regression models. Following a systematic search of Web of Science and PubMed, 55 records were identified that propose or evaluate such models. We describe the key models, summarise study characteristics, and present applications from biomedical and public health research. Clustering-based outcome models are particularly relevant for settings with high-dimensional covariates (e.g., biomarker panels and "omics") and heterogeneous patient populations. These models can support risk stratification and we discuss extensions to estimate subgroup-specific treatment effects. They are most valuable when the population is clustered in distinct regions of the covariate space that correspond to different outcome distributions. We discuss applications to rare disease research, covariate adjustment and borrowing from historical data, and subgroup-specific treatment effect estimation in clinical trials.

The recently developed rerandomized inverse variance weighted (RIVW) estimator provides a simple and efficient framework to break the winner's curse in two-sample Mendelian randomization (MR). However, this method does not account for sample structure (e.g., residual population stratification and sample overlap), a common source of confounding in MR studies. Sample structure can not only distort SNP-exposure and SNP-outcome association estimates but also induce correlation between them, leading exposure-side instrument selection to propagate bias to the outcome side. To address this challenge, we propose the bivariate RIVW (BRIVW) estimator to simultaneously account for the winner's curse and sample structure. The BRIVW estimator extends the RIVW framework by modeling the joint distribution of SNP-exposure and SNP-outcome association estimates, first adjusting their covariance matrix via linkage disequilibrium score regression to account for sample structure and then applying randomized instrument selection and bivariate Rao-Blackwellization to obtain unbiased post-selection association estimates together with an estimator of their covariance matrix. Under mild conditions, we show that the BRIVW estimator is consistent and asymptotically normal. The finite-sample performance of the proposed estimator is evaluated through extensive simulations and real data analyses.

We develop a hierarchical Bayesian dynamic game for competitive inventory and pricing under incomplete information. Two firms repeatedly choose order quantities and prices while facing two layers of uncertainty: unknown market demand and private rival characteristics. The framework combines Bayesian learning about demand and substitution with strategic belief updating about rival types. To make decisions robust to posterior uncertainty, we introduce a credible-risk criterion that rewards expected future profit while penalizing posterior predictive dispersion. This yields a conservative equilibrium concept in which firms learn, compete, and adapt simultaneously. The paper provides the model formulation, information structure, posterior updating mechanism, equilibrium definition, and a computational strategy based on belief-state dynamic programming. A simulation study shows that Bayesian learning is crucial for strong performance and that the credible-risk rule is especially effective as an operational regularizer under uncertainty. A real-data illustration on a high-dimensional protein-expression dataset demonstrates that the same uncertainty-aware Bayesian principle can produce biologically interpretable subgroup and latent-state findings. The proposed framework offers a unified bridge between Bayesian game theory and operations research, with practical relevance for competitive decision-making in uncertain and information-limited environments.

Continuous-time autoregressive and moving average (CARMA) models are extensively used to model high-frequency and irregularly sampled data. We study Whittle estimation for the model parameters when the process is observed at renewal times. The driving noise is assumed to be a Lévy process allowing for more flexibility including heavy-tailed marginal distributions and jumps in the sample paths. We show that the Whittle estimator based on the integrated periodogram is consistent and asymptotically normal under very mild conditions. To obtain these results, we establish the asymptotic normality of the integrated periodogram.

The complexity of learning a concept class under Gaussian marginals in the difficult agnostic model is closely related to its $L_1$-approximability by low-degree polynomials. For any concept class with Gaussian surface area at most $\Gamma$, Klivans et al. (2008) show that degree $d = O(\Gamma^2 / \varepsilon^4)$ suffices to achieve an $\varepsilon$-approximation. This leads to the best-known bounds on the complexity of learning a variety of concept classes. In this note, we improve their analysis by showing that degree $d = \tilde O (\Gamma^2 / \varepsilon^2)$ is enough. In light of lower bounds due to Diakonikolas et al. (2021), this yields (near) optimal bounds on the complexity of agnostically learning polynomial threshold functions in the statistical query model. Our proof relies on a direct analogue of a construction of Feldman et al. (2020), who considered $L_1$-approximation on the Boolean hypercube.

While suitably scaled CNNs with Gaussian initialization are known to converge to Gaussian processes as the number of channels diverges, little is known beyond this Gaussian limit. We establish a large deviation principle (LDP) for convolutional neural networks in the infinite-channel regime. We consider a broad class of multidimensional CNN architectures characterized by general receptive fields encoded through a patch-extractor function satisfying mild structural assumptions. Our main result establishes a large deviation principle (LDP) for the sequence of conditional covariance matrices under Gaussian prior distribution on the weights. We further derive an LDP for the posterior distribution obtained by conditioning on a finite number of observations. In addition, we provide a streamlined proof of the concentration of the conditional covariances and of the Gaussian equivalence of the network. To the best of our knowledge, this is the first large deviation principle established for convolutional neural networks.

Parameter estimation for the truncated skew-normal distribution is challenging, as truncation introduces additional nonlinearity into the likelihood function and often leads to numerical instability in existing estimation procedures. In this paper, we propose a grid-based estimation method, referred to as GRID-MOM, for parameter estimation in the truncated skew-normal distribution. The proposed approach fixes the shape parameter on a pre-specified grid and, for each grid point, estimates the location and scale parameters using the method of moments. The optimal value of the shape parameter is then selected via likelihood-based comparison, yielding the final parameter estimates. By decoupling the estimation of the shape parameter from that of the location and scale parameters, the proposed method reduces the complexity of the optimization problem and improves numerical stability. We evaluate the finite-sample performance of the proposed estimator through an extensive numerical study, comparing it with existing methods under a variety of scenarios. The results demonstrate that the proposed method provides stable and accurate estimation, particularly for the shape parameter, suggesting that the proposed method offers a practical alternative for inference in truncated skew-normal models. We further demonstrate the practical applicability of the proposed method using phosphoproteomics data and hospital admission data.

Gas gun and other shock compression experiments often produce shock wave velocity measurements that are linearly associated with particle velocity. Traditionally, this empirical relationship is quantified with a single Hugoniot curve that is estimated using least squares regression. However, for downstream modeling and simulation tasks, it is often more useful to have multiple Hugoniot curves in the pressure-volume plane that are consistent with the data. We employ Bayesian uncertainty quantification methods as a framework for propagating measurement uncertainty through to model parameters and predictions. Specifically, this tutorial shows how to sample multiple Hugoniot curves in the pressure-volume plane that are consistent with the shock wave-particle velocity measurements in a two-step Bayesian approach. First, we obtain an analytical expression for the posterior distribution of the linear model parameters using Bayesian linear regression. Second, we propagate samples from the posterior distribution through the Rankine-Hugoniot equations to yield Hugoniot curves in the pressure-volume plane. The procedure is demonstrated with publicly available data on argon, copper, and nickel, and compared against bootstrapping and linear regression. The Bayesian procedure is shown to be interpretable, computationally inexpensive, and less sensitive than an alternative bootstrapping approach to the removal of the point in the copper dataset that has the largest particle velocity. As a tutorial on Bayesian methodology for the shock compression community, we provide several derivations and explanations that make this paper self-contained, and made all code and data available at https://github.com/llnl/BALSCD.

Animal acoustic communication often exhibits temporal dependence, with calls triggering or suppressing subsequent calls within and across call types, individuals, or species. While Hawkes processes provide a natural framework for modeling excitation, incorporating inhibition in multivariate settings can raise identifiability issues and complicate parameter interpretation. We propose a flexible class of multivariate Hawkes processes that combines additive excitation with multiplicative inhibition. This formulation preserves the branching process interpretation of excitation while reducing confounding between excitation and inhibition, and allows direct quantification of background and excitation contributions to the event rate. Bayesian inference is conducted via Markov chain Monte Carlo, and model adequacy is assessed using the random time change theorem. The proposed methodology is evaluated through simulation and applied to two acoustic communication datasets: group-living meerkats, for which we analyze three selected call types with distinct behavioral roles, and a two-species baleen whale dataset involving humpback and North Atlantic right whales. The meerkat analysis reveals significant within- and cross-type excitation with cross-type inhibition, whereas the whale data show evidence primarily of within-species excitation.

Online platforms routinely compare multi-armed bandit algorithms, such as UCB and Thompson Sampling, to select the best-performing policy. Unlike standard A/B tests for static treatments, each run of a bandit algorithm over $T$ users produces only one trajectory, because the algorithm's decisions depend on all past interactions. Reliable inference therefore demands many independent restarts of the algorithm, making experimentation costly and delaying deployment decisions. We propose Artificial Replay (AR) as a new experimental design for this problem. AR first runs one policy and records its trajectory. When the second policy is executed, it reuses a recorded reward whenever it selects an action the first policy already took, and queries the real environment only otherwise. We develop a new analytical framework for this design and prove three key properties of the resulting estimator: it is unbiased; it requires only $T + o(T)$ user interactions instead of $2T$ for a run of the treatment and control policies, nearly halving the experimental cost when both policies have sub-linear regret; and its variance grows sub-linearly in $T$, whereas the estimator from a naïve design has a linearly-growing variance. Numerical experiments with UCB, Thompson Sampling, and $\epsilon$-greedy policies confirm these theoretical gains.

We consider nonparametric regression under covariate shift, where we observe samples from both the target distribution and a related but distinct source distribution. We introduce a novel object, the transfer function, and show that properties of its domain determine our minimax rates. Those exhibit a variety of regimes, including classical rates, governed by the better of source-only and target-only rates, as well as regimes in which the convergence rates exhibit multiplicative interactions between the sample sizes and are faster than the best-of-two benchmark. The rates are shown to be achieved up to logarithmic factors by a design-adaptive estimator. Compared with existing theory, our results cover the case in which covariates have unbounded support.

Linear programming is widely used for decision-making in science, engineering, and operations research, yet in many modern applications the coefficients entering the constraints and objective are not known exactly and must be learned from data. Classical stochastic and robust optimization offer two influential paradigms for handling such uncertainty, but they typically treat the underlying uncertainty description as given and do not directly integrate priors and updated to posteriors guarantees. This paper develops a Bayesian framework for linear programming in which uncertain quantities are modeled probabilistically, updated through observed data, and propagated into optimization through posterior feasibility requirements. We present two complementary computational strategies: a credible-region robustification that converts posterior uncertainty into deterministic protection, and a posterior-scenario approach that uses sampled posterior realizations to construct tractable optimization problems with finite-sample interpretability. We also propose a Monte Carlo certification procedure that provides conservative, data-conditioned assessments of residual infeasibility. Simulation experiments show that the proposed framework substantially improves safety relative to naive plug-in decisions, while a real-data study on single-cell transcriptomic data demonstrates that the approach can produce scientifically interpretable decisions together with explicit uncertainty-aware feasibility diagnostics. The proposed methodology offers a unified bridge between Bayesian learning, optimization under uncertainty, and practical decision certification.

We present a localized data assimilation (DA) scheme based on sequential Markov Chain Monte Carlo (SMCMC) \cite{smcmc}, a provably convergent method for filtering high-dimensional, nonlinear, and potentially non-Gaussian state-space models. Unlike particle filters, which are exact for such models, SMCMC does not assign weights to samples, avoiding weight degeneracy in small-ensemble regimes. We design two localization approaches within the SMCMC framework that exploit observation sparsity to reduce effective degrees of freedom. The first variant collects observed blocks into a single reduced domain and runs parallel MCMC chains over this combined region. The second further reduces per-chain state dimension by decomposing the observed region into independent blocks with compact halos, applying Gaspari–Cohn tapering to down-weight distant observations. When the observation model is linear Gaussian, the approximate filtering density reduces to a Gaussian mixture from which independent samples can be drawn exactly; otherwise, we employ an MCMC kernel. We test on high-dimensional ($d \sim 10^4 - 10^5$) state-space models, including a linear Gaussian model and a nonlinear multilayer shallow water equation with linear and nonlinear observation operators. We consider Gaussian and Student-$t$ observation noise, showing that LSMCMC handles heavy-tailed errors that cause ensemble Kalman methods to diverge. Observations include synthetic and real SWOT (NASA) and ocean drifter (NOAA) data. We compare the two variants against each other and the local ensemble transform Kalman filter (LETKF).

We propose a robust method for location estimation in various matrix manifolds based on the projected Frobenius median, which is closely related to the spatial median. This method applies broadly to matrix manifolds, including Stiefel and Grassmann manifolds, Kendall shape spaces as well as to projective Stiefel manifolds, a type of quotient space of a Stiefel manifold. Our approach involves computation of the Frobenius median in an ambient Euclidean space followed by projection onto the relevant matrix manifold. Our estimation method is computationally attractive, has a unique solution provided the sample data are not colinear in the ambient Euclidean space, has desirable robustness features and has appropriate equivariance properties under natural groups of transformations. We establish asymptotic normality under mild conditions and derive the influence function for matrix manifolds of interest. Simulation studies on the rank-1 complex Grassmann manifold and the projective Stiefel manifold further show the applicability and robustness of our method. We also apply our method to a real-world earthquake moment tensor dataset.

In the artificial intelligence (AI) age, firms increasingly invest in AI technology innovation to secure competitive advantages. However, the relationship between firms' AI technology innovation and consumer complaints remains insufficiently explored. Drawing on Protection Motivation Theory (PMT), this paper investigates how firms' AI technology innovation influences consumer complaints. Employing a multimethod approach, Study 1 analyzes panel data from S&P 500 firms (N = 2,758 firm-year observations), Study 2 examines user-generated Reddit data (N = 2,033,814 submissions and comments), and Study 3 involves two controlled experiments (N = 410 and N = 500). The results reveal that firms' AI technology innovation significantly increases consumers' threat-related emotions, heightening their complaints. Furthermore, compared to AI process innovation, AI product innovation leads to higher consumer complaints. This paper advances the understanding of consumers' psychological responses to firms' AI innovation and provides practical implications for managing consumer complaints effectively.

Frogeye Leaf Spot (FLS), caused by Cercospora sojina, poses a significant threat to soybean production, with yield losses of 30-60%. Traditional mass-action models assume homogeneous mixing, which rarely holds in real fields and limits their ability to inform FLS management. To address this, we developed a network-based model that incorporates real-field structure to improve FLS management in soybeans. Using approximate Bayesian computation, we estimated key epidemiological parameters and found that infection origin can shift the balance between transmission routes. Data analyses indicated that tillage and non-tillage plots did not differ significantly in fungal spread, decay, or disease severity. Finally, we show that early, targeted roguing is more effective than delayed or random removal. Together, these findings offer science-based guidance for FLS management and highlight the value of network-based models to inform agricultural disease control.

Total knee arthroplasty (TKA) and total hip arthroplasty (THA) improve symptoms in end-stage osteoarthritis, yet long-term objective characterization of perioperative physical activity trajectories remains limited. We conducted a longitudinal observational study within the All of Us Research Program dataset, linking electronic health records with continuous Fitbit-derived step count data over a four-year perioperative window (two years before and two years after arthroplasty). Piecewise linear mixed-effects models characterized preoperative declines and postoperative recovery trajectories, and time-to-recovery was evaluated using Kaplan-Meier curves and Cox proportional hazards models under remote and immediate preoperative physical activity baseline definitions. Among 238 participants (147 TKA; 91 THA), both procedures exhibited progressive preoperative decline with distinct procedure-specific patterns and staged postoperative recovery: rapid improvement during weeks 1-6, decelerating gains through weeks 7-19/20, and subsequent stabilization through week 104. Recovery to remote and immediate baselines differed in timing (median 22 vs 13 weeks) and associated predictors. Higher immediate preoperative activity was associated with greater likelihood of recovery to habitual activity levels, underscoring the relevance of preoperative functional reserve and surgical timing. These findings demonstrate the potential of long-term wearable monitoring to refine assessment of functional outcomes, guide recovery expectations, and support perioperative management.

Can we learn the differential equations governing the evolution of a temporal network? We investigate this within Random Dot Product Graphs (RDPGs), where each network snapshot is generated from latent positions evolving under unknown dynamics. We identify three fundamental obstructions: gauge freedom from rotational ambiguity in latent positions, realizability constraints from the manifold structure of the probability matrix, and trajectory recovery artifacts from spectral embedding. We develop a geometric framework based on principal fiber bundles that formalizes these obstructions. We characterize invisible dynamics as exactly the skew-symmetric generators, and show the realizable tangent space has dimension $nd - d(d-1)/2$. An holonomy dichotomy emerges: polynomial dynamics have commuting generators, stationary eigenvectors, and trivial holonomy, making gauge alignment purely statistical; Laplacian dynamics satisfy a non-commutativity criterion producing nontrivial holonomy, with curvature weighted by $1/(\lambda_\iota + \lambda_\gamma)$ linking gauge sensitivity to the spectral gap. In $d=2$ this yields full restricted holonomy $\mathrm{SO}(2)$; for $d \ge 3$ generic full $\mathrm{SO}(d)$ remains conjectural. Cram'er–Rao lower bounds reveal that the same spectral gap controlling curvature and injectivity simultaneously controls Fisher information, so geometric and statistical difficulty are inextricable. We prove an identifiability principle: symmetric dynamics cannot absorb skew-symmetric gauge contamination, so dynamics structure can resolve gauge ambiguity. We demonstrate this constructively with anchor-based alignment and a UDE pipeline recovering vector fields from noisy graph sequences. Yet finite-sample interactions between noise, gauge, and dynamics expressiveness remain beyond the asymptotic theory. We frame this gap as an open challenge.

The ocean is filled with phytoplankton that contribute as much photosynthesis as all land plants combined, making them vital to the carbon cycle and climate system. Recent advances in flow cytometry allow oceanographers to measure the optical traits of individual cells along research cruise tracks, generating single-cell resolution microbial data. In marine microbial ecology, detecting locations of abrupt changes in the environmental response of cytometric plankton distributions is an important task. This manuscript proposes a latent space Gaussian mixture-of-experts model, facilitating change point detection in replicated and clustered phytoplankton observations. Change points are identified through shifts in prior means of low-dimensional representations, with piecewise-constant structure enforced by a group-fused LASSO penalty. The optimization problem is then solved via Alternating Direction Method of Multipliers. Applied to flow cytometry data, the proposed method identifies a scientifically important change point that aligns with a transition zone between two marine provinces.

It is increasingly common in machine learning to use learned models to label data and then employ such data to train more capable models. The phenomenon of weak-to-strong generalization exemplifies the advantage of this two-stage procedure: a strong student is trained on imperfect labels obtained from a weak teacher, and yet the strong student outperforms the weak teacher. In this paper, we show that the potential improvement is substantial, in the sense that it affects the scaling law followed by the test error. Specifically, we consider students and teachers trained via random feature ridge regression (RFRR). Our main technical contribution is to derive a deterministic equivalent for the excess test error of the student trained on labels obtained via the teacher. Via this deterministic equivalent, we then identify regimes in which the scaling law of the student improves upon that of the teacher, unveiling that the improvement can be achieved both in bias-dominated and variance-dominated settings. Strikingly, the student may attain the minimax optimal rate regardless of the scaling law of the teacher – in fact, when the test error of the teacher does not even decay with the sample size.

We study discounted infinitely repeated games in which players agree on a cooperative mixed action profile but, at each step, observe only the realized pure actions. This form of imperfect monitoring breaks classical trigger strategies, since deviations cannot be identified with certainty. To address this problem, we study how hypothesis testing can be used to sustain cooperation. First, we develop a framework that embeds statistical inference directly into strategic behavior. We introduce relaxed equilibrium notions that allow players to ignore vanishing probability histories arising from rare but extreme realizations of the monitoring process. Within this framework, we formalize a generic test then punish strategy: players commit ex ante to a cooperative mixed action profile, continuously test whether observed play is consistent with this prescription, and permanently switch to punishment once sufficient statistical evidence of deviation accumulates. Under mild conditions on the testing procedure, this construction sustains any feasible and individually rational payoff for sufficiently patient players, yielding a Folk theorem type result under imperfect monitoring. We then propose two explicit implementations of this strategy. The first relies on anytime valid sequential tests, providing uniform control of Type I error over an infinite horizon and a finite expected detection time for payoff-relevant deviations. However, this strategy only accounts for stationary deviations and yields a Nash equilibrium. The second uses testing over batches with a fixed size, accommodating arbitrary deviations and achieving subgame perfect Nash equilibrium, at the cost of losing global anytime guarantees on false punishments.

Brain-wide recordings of large-scale networks of neurons now provide an unprecedented view into how the brain drives behavior. However, brain activity contains both information directly related to behavior as well as the potential for many internal computations. Moreover, observable behavior is executed not only by the brain, but also by the spinal cord and peripheral nervous system. Behavior is a coarse-grained product of neural activity, and we thus take the view that it can be best represented by lower-dimensional latent neural dynamics. Capturing this indirect relationship while disambiguating behavior-generating networks from internal computations running in parallel requires new modeling approaches that can embody the parallel and distributed nature of large-scale neural populations. We thus present behavior-decomposed linear dynamical systems (b-dLDS) to disentangle simultaneously recorded subsystems and identify how the latent neural subsystems relate to behavior. We demonstrate the ability of b-dLDS to decouple behavioral vs. internal computations on controlled, simulated data, showing improvements over a state-of-the-art model that uses behavior to supervise all dynamics based on behavior. We also demonstrate b-dLDS's interpretability benefits on a task-driven RNN dataset featuring a nonlinear relationship between behavior and activations. We then show that b-dLDS can further scale up to tens of thousands of neurons by applying our model to a large-scale recording of a zebrafish hindbrain during the complex positional homeostasis behavior, wherein b-dLDS highlights asymmetry in behavior-related dynamic connectivity networks.

We study prediction-powered conditional inference in the setting where labeled data are scarce, unlabeled covariates are abundant, and a black-box machine-learning predictor is available. The goal is to perform statistical inference on conditional functionals evaluated at a fixed target point, such as conditional means, without imposing a parametric model for the conditional relationship. Our approach combines localization with prediction-based variance reduction. First, we introduce an RKHS localization method that learns a data-adaptive weight from covariates and reformulates the target conditional moment at the target point as a weighted unconditional moment. Second, we incorporate machine-learning predictions through a correction-based decomposition of this localized moment, yielding a prediction-powered estimator and confidence interval that reduce variance when the predictor is informative while preserving validity regardless of predictor accuracy. We establish nonasymptotic error bounds and, in the abundant-unlabeled regime, minimax-optimal convergence rates for the resulting estimator, prove pointwise asymptotic normality with consistent variance estimation, and provide an explicit variance decomposition that characterizes how machine-learning predictions and unlabeled covariates improve statistical efficiency. Numerical experiments on simulated and real datasets demonstrate valid conditional coverage and substantially sharper confidence intervals than alternative methods.

Integrative analysis of multiple datasets for estimating optimal individualized treatment rules (ITRs) can enhance decision efficiency. A central challenge is posterior shift, wherein the conditional distribution of potential outcomes given covariates differs between source and target populations. We propose a prior information-based distributionally robust ITR (PDRO-ITR) that maximizes the worst-case policy value over a covariate-dependent distributional uncertainty set, ensuring robust performance under posterior shift. The uncertainty set is constructed as an individualized combination of source distributions, with weights combining prior source-membership probabilities and deviation terms constrained to the probability simplex to accommodate posterior shift. We derive a closed-form solution for the PDRO-ITR and develop an adaptive procedure to tune the uncertainty level. We establish risk bounds for the PDRO-ITR estimator, which guarantees robust performance under the worst case. Extensive simulations and two real-data applications demonstrate that the proposed method achieves superior performance compared to existing approaches.