Bayesian neural networks (BNNs) offer a natural probabilistic formulation for inference in deep learning models. Despite their popularity, their optimality has received limited attention through the lens of statistical decision theory. In this paper, we study decision rules induced by deep, fully connected feedforward ReLU BNNs in the normal location model under quadratic loss. We show that, for fixed prior scales, the induced Bayes decision rule is not minimax. We then propose a hyperprior on the effective output variance of the BNN prior that yields a superharmonic square-root marginal density, establishing that the resulting decision rule is simultaneously admissible and minimax. We further extend these results from the quadratic loss setting to the predictive density estimation problem with Kullback--Leibler loss. Finally, we validate our theoretical findings numerically through simulation.

Expectation Propagation (EP) is a widely used iterative message-passing algorithm that decomposes a global inference problem into multiple local ones. It approximates marginal distributions as ``beliefs'' using intermediate functions called ``messages''. It has been shown that the stationary points of EP are the same as corresponding constrained Bethe Free Energy (BFE) optimization problem. Therefore, EP is an iterative method of optimizing the constrained BFE. However, the iterative method may fall out of the feasible set of the BFE optimization problem, i.e., the beliefs are not integrable. In most literature, the authors use various methods to keep all the messages integrable. In most Bayesian estimation problems, limiting the messages to be integrable shrinks the actual feasible set. Furthermore, in extreme cases where the factors are not integrable, making the message itself integrable is not enough to have integrable beliefs. In this paper, two EP frameworks are proposed to ensure that EP has integrable beliefs. Both of the methods allows non-integrable messages. We then investigate the signal recovery problem in Generalized Linear Model (GLM) using our proposed methods.

When deploying large language models (LLMs) to safety-critical applications, uncertainty quantification (UQ) is of utmost importance to self-assess the reliability of the LLM-based decisions. However, such decisions typically suffer from overconfidence, particularly after parameter-efficient fine-tuning (PEFT) for downstream domain-specific tasks with limited data. Existing methods to alleviate this issue either rely on Laplace approximation based post-hoc framework, which may yield suboptimal calibration depending on the training trajectory, or variational Bayesian training that requires multiple complete forward passes through the entire LLM backbone at inference time for Monte Carlo estimation, posing scalability challenges for deployment. To address these limitations, we build on the Bayesian last layer (BLL) model, where the LLM-based deterministic feature extractor is followed by random last layer parameters for uncertainty reasoning. Since existing low-rank adapters (LoRA) for PEFT have limited expressiveness due to rank collapse, we address this with Polar-decomposed Low-rank Adapter Representation (PoLAR), an orthogonalized parameterization paired with Riemannian optimization to enable more stable and expressive adaptation. Building on this PoLAR-BLL model, we leverage the variational (V) inference framework to put forth a scalable Bayesian fine-tuning approach which jointly seeks the PoLAR parameters and approximate posterior of the last layer parameters via alternating optimization. The resulting PoLAR-VBLL is a flexible framework that nicely integrates architecture-enhanced optimization with scalable Bayesian inference to endow LLMs with well-calibrated UQ. Our empirical results verify the effectiveness of PoLAR-VBLL in terms of generalization and uncertainty estimation on both in-distribution and out-of-distribution data for various common-sense reasoning tasks.

Estimating optimal individualized treatment rules (ITRs) via outcome weighted learning (OWL) often relies on observed rewards that are noisy or optimistic proxies for the true latent utility. Ignoring this reward uncertainty leads to the selection of policies with inflated apparent performance, yet existing OWL frameworks lack the finite-sample guarantees required to systematically embed such uncertainty into the learning objective. To address this issue, we propose PAC-Bayesian Reward-Certified Outcome Weighted Learning (PROWL). Given a one-sided uncertainty certificate, PROWL constructs a conservative reward and a strictly policy-dependent lower bound on the true expected value. Theoretically, we prove an exact certified reduction that transforms robust policy learning into a unified, split-free cost-sensitive classification task. This formulation enables the derivation of a nonasymptotic PAC-Bayes lower bound for randomized ITRs, where we establish that the optimal posterior maximizing this bound is exactly characterized by a general Bayes update. To overcome the learning-rate selection problem inherent in generalized Bayesian inference, we introduce a fully automated, bounds-based calibration procedure, coupled with a Fisher-consistent certified hinge surrogate for efficient optimization. Our experiments demonstrate that PROWL achieves improvements in estimating robust, high-value treatment regimes under severe reward uncertainty compared to standard methods for ITR estimation.

We study adversarial learning when the target distribution factorizes according to a known Bayesian network. For interpolative divergences, including $(f,Γ)$-divergences, we prove a new infimal subadditivity principle showing that, under suitable conditions, a global variational discrepancy is controlled by an average of family-level discrepancies aligned with the graph. In an additive regime, the surrogate is exact. This closes a theoretical gap in the literature; existing subadditivity results justify graph-informed adversarial learning for classical discrepancies, but not for interpolative divergences, where the usual factorization argument breaks down. In turn, we provide a justification for replacing a standard, graph-agnostic GAN with a monolithic discriminator by a graph-informed GAN (GiGAN) with localized family-level discriminators, without requiring the optimizer itself to factorize according to the graph. We also obtain parallel results for integral probability metrics and proximal optimal transport divergences, identify natural discriminator classes for which the theory applies, and present experiments showing improved stability and structural recovery relative to graph-agnostic baselines.

Modern real-time systems require accurate characterization of task timing behavior to ensure predictable performance, particularly on complex hardware architectures. Existing methods, such as worst-case execution time analysis, often fail to capture the fine-grained timing behaviors of a task under varying resource contexts (e.g., an allocation of cache, memory bandwidth, and CPU frequency), which is necessary to achieve efficient resource utilization. In this paper, we introduce a novel generative profiling approach that synthesizes context-dependent, fine-grained timing profiles for real-time tasks, including those for unmeasured resource allocations. Our approach leverages a nonparametric, conditional multi-marginal Schrödinger Bridge (MSB) formulation to generate accurate execution profiles for unseen resource contexts, with maximum likelihood guarantees. We demonstrate the efficiency and effectiveness of our approach through real-world benchmarks, and showcase its practical utility in a representative case study of adaptive multicore resource allocation for real-time systems.

We propose closed-form conditional diffusion models for data assimilation. Diffusion models use data to learn the score function (defined as the gradient of the log-probability density of a data distribution), allowing them to generate new samples from the data distribution by reversing a noise injection process. While it is common to train neural networks to approximate the score function, we leverage the analytical tractability of the score function to assimilate the states of a system with measurements. To enable the efficient evaluation of the score function, we use kernel density estimation to model the joint distribution of the states and their corresponding measurements. The proposed approach also inherits the capability of conditional diffusion models of operating in black-box settings, i.e., the proposed data assimilation approach can accommodate systems and measurement processes without their explicit knowledge. The ability to accommodate black-box systems combined with the superior capabilities of diffusion models in approximating complex, non-Gaussian probability distributions means that the proposed approach offers advantages over many widely used filtering methods. We evaluate the proposed method on nonlinear data assimilation problems based on the Lorenz-63 and Lorenz-96 systems of moderate dimensionality and nonlinear measurement models. Results show the proposed approach outperforms the widely used ensemble Kalman and particle filters when small to moderate ensemble sizes are used.

We review recently proposed Bayesian approaches for clustering high-dimensional data. After identifying the main limitations of available approaches, we introduce an alternative framework based on vertical consensus inference (VCI) to mitigate the curse of dimensionality in high-dimensional Bayesian clustering. VCI builds on the idea of consensus Monte Carlo by dividing the data into multiple shards (smaller subsets of variables), performing posterior inference on each shard, and then combining the shard-level posteriors to obtain a consensus posterior. The key distinction is that VCI splits the data vertically, producing vertical shards that retain the same number of observations but have lower dimensionality. We use an entropic regularized Wasserstein barycenter to define a consensus posterior. The shard-specific barycenter weights are constructed to favor shards that provide meaningful partitions, distinct from a trivial single cluster or all singleton clusters, favoring balanced cluster sizes and precise shard-specific posterior random partitions. We show that VCI can be interpreted as a variational approximation to the posterior under a hierarchical model with a generalized Bayes prior. For relatively low-dimensional problems, experiments suggest that VCI closely approximates inference based on clustering the entire multivariate data. For high-dimensional data and in the presence of many noninformative dimensions, VCI introduces a new framework for model-based and principled inference on random partitions. Although our focus here is on random partitions, VCI can be applied to any dimension-independent parameters and serves as a bridge to emerging areas in statistics such as consensus Monte Carlo, optimal transport, variational inference, and generalized Bayes.

Preference-based many-objective optimization faces two obstacles: an expanding space of trade-offs and heterogeneous, context-dependent human value structures. Towards this, we propose a Bayesian framework that learns a small set of latent preference archetypes rather than assuming a single fixed utility function, modelling them as components of a Dirichlet-process mixture with uncertainty over both archetypes and their weights. To query efficiently, we designing hybrid queries that target information about (i) mode identity and (ii) within-mode trade-offs. Under mild assumptions, we provide a simple regret guarantee for the resulting mixture-aware Bayesian optimization procedure. Empirically, our method outperforms standard baselines on synthetic and real-world many-objective benchmarks, and mixture-aware diagnostics reveal structure that regret alone fails to capture.

We introduce a novel class of graphical models, termed profile graphical models, that represent, within a single graph, how an external factor influences the dependence structure of a multivariate set of variables. This class is quite general and includes multiple graphs and chain graphs as special cases. Profile graphical models capture the conditional distributions of a multivariate random vector given different levels of a risk factor, and learn how the conditional independence structure among variables may vary across these risk profiles; we formally define this family of models and establish their corresponding Markov properties. We derive key structural and probabilistic properties that underpin a more powerful inferential framework than existing approaches, underscoring that our contribution extends beyond a novel graphical representation.Furthermore, we show that the resulting profile undirected graphical models are independence-compatible with two-block LWF chain graph models.We then develop a Bayesian approach for Gaussian undirected profile graphical models based on continuous spike-and-slab priors to learn shared sparsity structures across different levels of the risk factor. We also design a fast EM algorithm for efficient inference. Inferential properties are explored through simulation studies, including the comparison with competing methods. The practical utility of this class of models is demonstrated through the analysis of protein network data from various subtypes of acute myeloid leukemia. Our results show a more parsimonious network and greater patient heterogeneity than its competitors, highlighting its enhanced ability to capture subject-specific differences.