However, deploying these models can be unreliable when significant distribution gaps exist between training and test data, while fine-tuning for diverse scenarios is often costly. This creates a need for methods that can efficiently adapt to new data at test time without expensive retraining. Cache-based test-time adapters serve this purpose by storing representative test samples to guide subsequent classifications. Yet, these methods typically employ naive cache management with limited capacity, leading to severe catastrophic forgetting when samples are inevitably dropped during updates. In this paper, we propose Dota(DistributiOnal Test-time Adaptation), a simple yet effective method addressing this limitation. Crucially, instead of merely memorizing individual test samples, Dotacontinuously estimates the underlying distribution of the test data stream. Test-time posterior probabilities are then computed using these dynamically estimated distributions via Bayes' theorem for adaptation. This distribution-centric approach enables the model to continually learn and adapt to the deployment environment. Extensive experiments validate that Dota significantly mitigates forgetting and achieves state-of-the-art performance compared to existing methods.

Targeted maximum likelihood estimation (TMLE) is a widely used debiasing algorithm for plug-in estimation. While its statistical guarantees, such as double robustness and asymptotic efficiency, are well-studied, the convergence properties of TMLE as an iterative optimization scheme have remained underexplored. To bridge this gap, we study TMLE's iterative updates through an optimization-theoretic lens, establishing global convergence under standard assumptions and regularity conditions. We begin by providing the first complete characterization of different stopping criteria and their relationship to convergence in TMLE. Next, we provide geometric insights.

For continuing tasks, average cost Markov decision processes have welldocumented value and can be solved using efficient algorithms. However, it explicitly assumes that the agent is risk-neutral. In this work, we extend risk-neutral algorithms to accommodate the more general class of dynamic risk measures. Specifically, we propose a relative value iteration (RVI) algorithm for planning and design two model-free Q-learning algorithms, namely a generic algorithm based on the multi-level Monte Carlo (MLMC) method, and an off-policy algorithm dedicated to utility-based shortfall risk measures. Both the RVI and MLMC-based Qlearning algorithms are proven to converge to optimality. Numerical experiments validate our analysis, confirm empirically the convergence of the off-policy algorithm, and demonstrate that our approach enables the identification of policies that are finely tuned to the intricate risk-awareness of the agent that they serve.

Bayesian networks (BNs), represented by directed acyclic graphs (DAGs), provide a principled framework for modeling complex dependencies among random variables. As data dimensionality increases into the tens of thousands, fitting and marginalizing a full BN becomes computationally prohibitive--particularly when inference is only needed for a small subset of variables. Estimation-collapsibility addresses this challenge by ensuring that directly fitting a submodel, obtained by ignoring non-essential variables, still yields exact inference on target variables. However, current DAG-based criterion for checking estimation-collapsibility is computationally intensive, involving exhaustive vertex searches and iterative removals. Additionally, practical applications typically identify the underlying DAG only up to its Markov equivalence class, represented by a completed partially directed acyclic graph (CPDAG). To bridge this gap, we introduce sequential c-simplicial sets--a novel graphical characterization of estimation-collapsibility directly applicable to CPDAGs. We further propose DSCS, a computationally efficient algorithm for verifying estimation-collapsibility within CPDAG framework that scales effectively to high-dimensional BNs. Extensive numerical experiments demonstrate the practicality, scalability, and efficiency of our proposed approach.

Online advertising platforms use automated auctions to connect advertisers with potential customers, requiring effective bidding strategies to maximize profits. Accurate ad impact estimation requires considering three key factors: delayed and long-term effects, cumulative ad impacts such as reinforcement or fatigue, and customer heterogeneity. However, these effects are often not jointly addressed in previous studies. To capture these factors, we model ad bidding as a Contextual Markov Decision Process (CMDP) with delayed Poisson rewards. For efficient estimation, we propose a two-stage maximum likelihood estimator combined with data-splitting strategies, ensuring controlled estimation error based on the first-stage estimator's (in)accuracy. Building on this, we design a reinforcement learning algorithm to derive efficient personalized bidding strategies. This approach achieves a near-optimal regret bound of O(dH2 T), where d is the contextual dimension, H is the number of rounds, and T is the number of customers. Our theoretical findings are validated by simulation experiments.

Quantifying differences between probability distributions is fundamental to statistics and machine learning, primarily for comparing statistical uncertainty. In contrast, epistemic uncertainty--due to incomplete knowledge--requires richer representations than those offered by classical probability. Imprecise probability (IP) theory offers such models, capturing ambiguity and partial belief. This has driven growing interest in imprecise probabilistic machine learning (IPML), where inference and decision-making rely on broader uncertainty models--highlighting the need for metrics beyond classical probability. This work introduces the Integral Imprecise Probability Metric (IIPM) framework, a Choquet integral-based generalisation of classical Integral Probability Metrics (IPMs) to the setting of capacities--a broad class of IP models encompassing many existing ones, including lower probabilities, probability intervals, belief functions, and more. Theoretically, we establish conditions under which IIPM serves as a valid metric and metrises a form of weak convergence of capacities. Practically, IIPM not only enables comparison across different IP models but also supports the quantification of epistemic uncertainty (EU) within a single IP model. In particular, by comparing an IP model with its conjugate, IIPM gives rise to a new class of EU measures--Maximum Mean Imprecisions (MMIs)--which satisfy key axiomatic properties proposed in the uncertainty quantification literature. We validate MMI through selective classification experiments, demonstrating strong empirical performance against established EU measures, and outperforming them when classical methods struggle to scale to a large number of classes.

Autoregressive generative models naturally generate variable-length sequences, while non-autoregressive models struggle, often imposing rigid, token-wise structures. We propose Edit Flows, a non-autoregressive model that overcomes these limitations by defining a discrete flow over sequences through edit operations-- insertions, deletions, and substitutions. By modeling these operations within a Continuous-time Markov Chain over the sequence space, Edit Flows enable flexible, position-relative generation that aligns more closely with the structure of sequence data. Our training method leverages an expanded state space with auxiliary variables, making the learning process efficient and tractable. Empirical results show that Edit Flows outperforms both autoregressive and mask models on image captioning and significantly outperforms the mask construction in text and code generation.

A generalizable reward model is crucial in Reinforcement Learning from Human Feedback (RLHF) as it enables correctly evaluating unseen prompt-response pairs. However, existing reward models can lack this ability, as they are typically trained by increasing the reward gap between the chosen and rejected responses, while overlooking the prompts that the responses are conditioned on. Consequently, when the trained reward model is evaluated on prompt-response pairs that lie outside the data distribution, neglecting the effect of prompts may result in poor generalization of the reward model. To address this issue, we decompose the reward value into two independent components: prompt-free reward and prompt-related reward. Prompt-free reward represents the evaluation that is determined only by responses, while the prompt-related reward reflects the reward that derives from both the prompt and the response. We extract these two components from an information-theoretic perspective, which requires no extra models. Subsequently, we propose a new reward learning algorithm by prioritizing data samples based on their prompt-free reward values. Through toy examples, we demonstrate that the extracted prompt-free and prompt-related rewards effectively characterize the two parts of the reward value. Further, standard evaluations show that our method improves both the alignment performance and the generalization capability of the reward model.

Recent advances in reasoning techniques have substantially improved the performance of large language models (LLMs), raising expectations for their ability to provide accurate, truthful, and reliable information. However, emerging evidence suggests that iterative reasoning may foster belief entrenchment, rather than enhancing truth-seeking behavior. In this study, we propose a systematic evaluation framework for belief entrenchment in LLM reasoning by leveraging the Martingale property from Bayesian statistics. This property implies that, under rational belief updating, the expected value of future beliefs should remain equal to the current belief, i.e., belief updates cannot be predicted from solely the current belief. We propose the unsupervised, regression-based Martingale Score to measure violations of this property, signaling a deviation from the Bayesian ability of updating on new evidence. In open-ended problem domains, including event forecasting, value-laden questions, and academic paper review, we found such violations to be widespread across models, reasoning paradigms, problem domains, and system prompts, where the future beliefs are consistently predictable from the model's current belief, a phenomenon which we term belief entrenchment. Through comprehensive experiments, we identify the models (e.g., GPT-4o), reasoning techniques (e.g., chain of thought), and domains (e.g., forecasting) more prone to belief entrenchment. Finally, we validate the Martingale Score by showing that it predicts ground-truth accuracy on problem domains where ground truth labels are available. This indicates that, while designed as an unsupervised metric that operates even in domains without access to ground truth, the Martingale Score is a useful proxy of the truth-seeking ability of the LLM reasoning process.

We introduce DeCaFlow, a deconfounding causal generative model. Training once per dataset using just observational data and the underlying causal graph, DeCaFlow enables accurate causal inference on continuous variables under the presence of hidden confounders. Specifically, we extend previous results on causal estimation under hidden confounding to show that a single instance of DeCaFlow provides correct estimates for all causal queries identifiable with do-calculus, leveraging proxy variables to adjust for the causal effects when do-calculus alone is insufficient. Moreover, we show that counterfactual queries are identifiable as long as their interventional counterparts are identifiable, and thus are also correctly estimated by DeCaFlow. Our empirical results on diverse settings--including the Ecoli70 dataset, with 3 independent hidden confounders, tens of observed variables and hundreds of causal queries--show that DeCaFlow outperforms existing approaches, while demonstrating its out-of-the-box applicability to any given causal graph.