In this paper, a Bayesian nonparametric (BNP) model for Baxter permutations (BPs), termed BP process (BPP) is proposed and applied to relational data analysis. The BPs are a well-studied class of permutations, and it has been demonstrated that there is one-to-one correspondence between BPs and several interesting objects including floorplan partitioning (FP), which constitutes a subset of rectangular partitioning (RP). Accordingly, the BPP can be used as an FP model. We combine the BPP with a multi-dimensional extension of the stick-breaking process called the block-breaking process to fill the gap between FP and RP, and obtain a stochastic process on arbitrary RPs. Compared with conventional BNP models for arbitrary RPs, the proposed model is simpler and has a high affinity with Bayesian inference.

In many scientific research and engineering applications where repeated simulations of complex systems are conducted, a surrogate is commonly adopted to quickly estimate the whole system. To reduce the expensive cost of generating training examples, it has become a promising approach to combine the results of low-fidelity (fast but inaccurate) and high-fidelity (slow but accurate) simulations. Despite the fast developments of multi-fidelity fusion techniques, most existing methods require particular data structures and do not scale well to high-dimensional output. To resolve these issues, we generalize the classic autoregression (AR), which is wildly used due to its simplicity, robustness, accuracy, and tractability, and propose generalized autoregression (GAR) using tensor formulation and latent features. GAR can deal with arbitrary dimensional outputs and arbitrary multifidelity data structure to satisfy the demand of multi-fidelity fusion for complex problems; it admits a fully tractable likelihood and posterior requiring no approximate inference and scales well to high-dimensional problems.

The overdamped Langevin stochastic differential equation (SDE) is a classical physical model used for chemical, genetic, and hydrological dynamics. In this work, we prove that the drift and diffusion terms of a Langevin SDE are jointly identifiable from temporal marginal distributions if and only if the process is observed out of equilibrium. This complete characterization of structural identifiability removes the long-standing assumption that the diffusion must be known to identify the drift. We then complement our theory with experiments in the finite sample setting and study the practical identifiability of the drift and diffusion, in order to propose heuristics for optimal data collection.

This paper proposes a framework for semantic hypothesis testing tailored to imaging inverse problems. Modern imaging methods struggle to support hypothesis testing, a core component of the scientific method that is essential for the rigorous interpretation of experiments and robust interfacing with decision-making processes. There are three main reasons why image-based hypothesis testing is challenging. First, the difficulty of using a single observation to simultaneously reconstruct an image, formulate hypotheses, and quantify their statistical significance. Second, the hypotheses encountered in imaging are mostly of semantic nature, rather than quantitative statements about pixel values. Third, it is challenging to control test error probabilities because the null and alternative distributions are often unknown. Our proposed approach addresses these difficulties by leveraging concepts from self-supervised computational imaging, vision-language models, and non-parametric hypothesis testing with e-values. We demonstrate our proposed framework through numerical experiments related to image-based phenotyping, where we achieve excellent power while robustly controlling Type I errors.

Data Augmentation (DA) has become an essential tool to improve robustness and generalization of modern machine learning. However, when deciding on DA strategies it is critical to choose parameters carefully, and this can be a daunting task which is traditionally left to trial-and-error or expensive optimization based on validation performance. In this paper, we counter these limitations by proposing a novel framework for optimizing DA. In particular, we take a probabilistic view of DA, which leads to the interpretation of augmentation parameters as model (hyper)-parameters, and the optimization of the marginal likelihood with respect to these parameters as a Bayesian model selection problem. Due to its intractability, we derive a tractable Evidence Lower BOund (ELBO), which allows us to optimize augmentation parameters jointly with model parameters. We provide extensive theoretical results on variational approximation quality, generalization guarantees, invariance properties, and connections to empirical Bayes. Through experiments on computer vision tasks, we show that our approach improves calibration and yields robust performance over fixed or no augmentation. Our work provides a rigorous foundation for optimizing DA through Bayesian principles with significant potential for robust machine learning.

Out-of-distribution (OOD) generalization remains a fundamental challenge in machine learning. Models trained on one data distribution often experience substantial performance degradation when evaluated on shifted or unseen domains. To address this challenge, the present paper studies the design of training data distributions that maximize average-case OOD performance. First, a theoretical analysis establishes a family of generalization bounds that quantify how the choice of training distribution influences OOD error across a predefined family of target distributions. These insights motivate the introduction of two complementary algorithmic strategies: (i) directly formulating OOD risk minimization as a bilevel optimization problem over the space of probability measures and (ii) minimizing a theoretical upper bound on OOD error. Last, the paper evaluates the two approaches across a range of function approximation and operator learning examples. The proposed methods significantly improve OOD accuracy over standard empirical risk minimization with a fixed distribution. These results highlight the potential of distribution-aware training as a principled and practical framework for robust OOD generalization.

Predictions in the form of sets of probability distributions, so-called credal sets, provide a suitable means to represent a learner's epistemic uncertainty. In this paper, we propose a theoretically grounded approach to credal prediction based on the statistical notion of relative likelihood: The target of prediction is the set of all (conditional) probability distributions produced by the collection of plausible models, namely those models whose relative likelihood exceeds a specified threshold. This threshold has an intuitive interpretation and allows for controlling the trade-off between correctness and precision of credal predictions. We tackle the problem of approximating credal sets defined in this way by means of suitably modified ensemble learning techniques. To validate our approach, we illustrate its effectiveness by experiments on benchmark datasets demonstrating superior uncertainty representation without compromising predictive performance. We also compare our method against several state-of-the-art baselines in credal prediction.

Automated evaluation of free-form outputs from large language models (LLMs) is challenging because many distinct answers can be equally valid. A common practice is to use LLMs themselves as judges, but the theoretical properties of this approach are not yet well understood. We show that a geometric framework that represents both judges and candidates as points on a probability simplex can provide helpful insight on what is or is not identifiable using LLM judges. Our theoretical analysis uncovers a "phase transition" in ranking identifiability: for binary scoring systems, true rankings are identifiable even with weak judges under mild assumptions, while rankings become non-identifiable for three or more scoring levels even with infinite data, absent additional prior knowledge. This non-identifiability highlights how uncertainty in rankings stems from not only aleatoric uncertainty (i.e., inherent stochasticity in the data) but also epistemic uncertainty regarding which assumptions hold, an aspect that has received limited attention until now. To integrate both types of uncertainty, we use Bayesian inference to encode assumptions as priors and conduct sensitivity analysis of ranking estimates and credible intervals. Empirical evaluations across multiple benchmarks demonstrate that Bayesian inference yields more accurate rankings and substantially improves coverage rates. These results underscore the importance of taking a more holistic approach to uncertainty quantification when using LLMs as judges.

We address the problem of uncertainty quantification and propose measures of total, aleatoric, and epistemic uncertainty based on a known decomposition of (strictly) proper scoring rules, a specific type of loss function, into a divergence and an entropy component. This leads to a flexible framework for uncertainty quantification that can be instantiated with different losses (scoring rules), which makes it possible to tailor uncertainty quantification to the use case at hand. We show that this flexibility is indeed advantageous. In particular, we analyze the task of selective prediction and show that the scoring rule should ideally match the task loss. In addition, we perform experiments on two other common tasks. For out-of-distribution detection, our results confirm that a widely used measure of epistemic uncertainty, mutual information, performs best. Moreover, in the setting of active learning, our measure of epistemic uncertainty based on the zero-one-loss consistently outperforms other uncertainty measures.

Bounded continuous responses -- such as proportions -- arise frequently in diverse scientific fields including climatology, biostatistics, and finance. Beta regression is a widely adopted framework for modeling such data, due to the flexibility of the Beta distribution over the unit interval. While Bayesian extensions of Beta regression have shown promise, existing methods are limited to low-dimensional settings and lack theoretical guarantees. In this work, we propose a novel Bayesian approach for high-dimensional sparse Beta regression framework that employs a tempered posterior. Our method incorporates the Horseshoe prior for effective shrinkage and variable selection. Most notable, we propose a novel Gibbs sampling algorithm using Pólya-Gamma augmentation for efficient inference in Beta regression model. We also provide the first theoretical results establishing posterior consistency and convergence rates for Bayesian Beta regression. Through extensive simulation studies in both low- and high-dimensional scenarios, we demonstrate that our approach outperforms existing alternatives, offering improved estimation accuracy and model interpretability. Our method is implemented in the R package ``betaregbayes" available on Github.