While offering a principled framework for uncertainty quantification in deep learning, the employment of Bayesian Neural Networks (BNNs) is still constrained by their increased computational requirements and the convergence difficulties when training very deep, state-of-the-art architectures. In this work, we reinterpret weight-sharing quantization techniques from a stochastic perspective in the context of training and inference with Bayesian Neural Networks (BNNs). Specifically, we leverage 2D adaptive Gaussian distributions, Wasserstein distance estimations, and alpha blending to encode the stochastic behaviour of a BNN in a lower dimensional, soft Gaussian representation. Through extensive empirical investigation, we demonstrate that our approach significantly reduces the computational overhead inherent in Bayesian learning by several orders of magnitude, enabling the efficient Bayesian training of large-scale models, such as ResNet-101 and Vision Transformer (VIT). On various computer vision benchmarks including CIFAR10, CIFAR100, and ImageNet1k. Our approach compresses model parameters by approximately 50x and reduces model size by 75, while achieving accuracy and uncertainty estimations comparable to the state-of-the-art.

Ill-posed inverse problems are fundamental in many domains, ranging from astrophysics to medical imaging. Emerging diffusion models provide a powerful prior for solving these problems. Existing maximum-a-posteriori (MAP) or posterior sampling approaches, however, rely on different computational approximations, leading to inaccurate or suboptimal samples. To address this issue, we introduce a new approach to solving MAP problems with diffusion model priors using a dual ascent optimization framework. Our framework achieves better image quality as measured by various metrics for image restoration problems, it is more robust to high levels of measurement noise, it is faster, and it estimates solutions that represent the observations more faithfully than the state of the art.

We propose a general framework for conditional sampling in PDE-based inverse problems, targeting the recovery of whole solutions from extremely sparse or noisy measurements. This is accomplished by a function-space diffusion model and plug-and-play guidance for conditioning. Our method first trains an unconditional discretization-agnostic denoising model using neural operator architectures. At inference, we refine the samples to satisfy sparse observation data via a gradient-based guidance mechanism. Through rigorous mathematical analysis, we extend Tweedie's formula to infinite-dimensional Hilbert spaces, providing the theoretical foundation for our posterior sampling approach. Our method (FunDPS) accurately captures posterior distributions in function spaces under minimal supervision and severe data scarcity. Across five PDE tasks with only 3% observation, our method achieves an average 32% accuracy improvement over state-of-the-art fixed-resolution diffusion baselines while reducing sampling steps by 4x. Furthermore, multi-resolution fine-tuning ensures strong cross-resolution generalizability. To the best of our knowledge, this is the first diffusion-based framework to operate independently of discretization, offering a practical and flexible solution for forward and inverse problems in the context of PDEs. Code is available at https://github.com/neuraloperator/FunDPS

Topic models often fail to capture low-prevalence, domain-critical themes, so-called minority topics, such as mental health themes in online comments. While some existing methods can incorporate domain knowledge, such as expected topical content, methods allowing guidance may require overly detailed expected topics, hindering the discovery of topic divisions and variation. We propose a topic modeling solution via a specially constrained NMF. We incorporate a seed word list characterizing minority content of interest, but we do not require experts to pre-specify their division across minority topics. Through prevalence constraints on minority topics and seed word content across topics, we learn distinct data-driven minority topics as well as majority topics. The constrained NMF is fitted via Karush-Kuhn-Tucker (KKT) conditions with multiplicative updates. We outperform several baselines on synthetic data in terms of topic purity, normalized mutual information, and also evaluate topic quality using Jensen-Shannon divergence (JSD). We conduct a case study on YouTube vlog comments, analyzing viewer discussion of mental health content; our model successfully identifies and reveals this domain-relevant minority content.

The power prior is a class of informative priors designed to incorporate historical data alongside current data in a Bayesian framework. It includes a power parameter that controls the influence of historical data, providing flexibility and adaptability. A key property of the power prior is that the resulting posterior minimizes a linear combination of KL divergences between two pseudo-posterior distributions: one ignoring historical data and the other fully incorporating it. We extend this framework by identifying the posterior distribution as the minimizer of a linear combination of Amari's $α$-divergence, a generalization of KL divergence. We show that this generalization can lead to improved performance by allowing for the data to adapt to appropriate choices of the $α$ parameter. Theoretical properties of this generalized power posterior are established, including behavior as a generalized geodesic on the Riemannian manifold of probability distributions, offering novel insights into its geometric interpretation.

Contextual online decision-making problems with constraints appear in a wide range of real-world applications, such as adaptive experimental design under safety constraints, personalized recommendation with resource limits, and dynamic pricing under fairness requirements. In this paper, we investigate a general formulation of sequential decision-making with stage-wise feasibility constraints, where at each round, the learner must select an action based on observed context while ensuring that a problem-specific feasibility criterion is satisfied. We propose a unified algorithmic framework that captures many existing constrained learning problems, including constrained bandits, active learning with label budgets, online hypothesis testing with Type I error control, and model calibration. Central to our approach is the concept of upper counterfactual confidence bounds, which enables the design of practically efficient online algorithms with strong theoretical guarantees using any offline conditional density estimation oracle. To handle feasibility constraints in complex environments, we introduce a generalized notion of the eluder dimension, extending it from the classical setting based on square loss to a broader class of metric-like probability divergences. This allows us to capture the complexity of various density function classes and characterize the utility regret incurred due to feasibility constraint uncertainty. Our result offers a principled foundation for constrained sequential decision-making in both theory and practice.

We study the problem of estimating a distribution over a finite alphabet from an i.i.d. sample, with accuracy measured in relative entropy (Kullback-Leibler divergence). While optimal expected risk bounds are known, high-probability guarantees remain less well-understood. First, we analyze the classical Laplace (add-one) estimator, obtaining matching upper and lower bounds on its performance and showing its optimality among confidence-independent estimators. We then characterize the minimax-optimal high-probability risk, which is attained via a simple confidence-dependent smoothing technique. Interestingly, the optimal non-asymptotic risk exhibits an additional logarithmic factor over the ideal asymptotic risk. Next, motivated by scenarios where the alphabet exceeds the sample size, we investigate methods that adapt to the sparsity of the distribution at hand. We introduce an estimator using data-dependent smoothing, for which we establish a high-probability risk bound depending on two effective sparsity parameters. As part of the analysis, we also derive a sharp high-probability upper bound on the missing mass.

Label shift adaptation aims to recover target class priors when the labelled source distribution $P$ and the unlabelled target distribution $Q$ share $P(X \mid Y) = Q(X \mid Y)$ but $P(Y) \neq Q(Y)$. Classical black-box shift estimators invert an empirical confusion matrix of a frozen classifier, producing a brittle point estimate that ignores sampling noise and similarity among classes. We present Graph-Smoothed Bayesian BBSE (GS-B$^3$SE), a fully probabilistic alternative that places Laplacian-Gaussian priors on both target log-priors and confusion-matrix columns, tying them together on a label-similarity graph. The resulting posterior is tractable with HMC or a fast block Newton-CG scheme. We prove identifiability, $N^{-1/2}$ contraction, variance bounds that shrink with the graph's algebraic connectivity, and robustness to Laplacian misspecification. We also reinterpret GS-B$^3$SE through information geometry, showing that it generalizes existing shift estimators.

Optimal decision-making under partial observability requires agents to balance reducing uncertainty (exploration) against pursuing immediate objectives (exploitation). In this paper, we introduce a novel policy optimization framework for continuous partially observable Markov decision processes (POMDPs) that explicitly addresses this challenge. Our method casts policy learning as probabilistic inference in a non-Markovian Feynman--Kac model that inherently captures the value of information gathering by anticipating future observations, without requiring extrinsic exploration bonuses or handcrafted heuristics. To optimize policies under this model, we develop a nested sequential Monte Carlo~(SMC) algorithm that efficiently estimates a history-dependent policy gradient under samples from the optimal trajectory distribution induced by the POMDP. We demonstrate the effectiveness of our algorithm across standard continuous POMDP benchmarks, where existing methods struggle to act under uncertainty.

We explore the use of conformal prediction to provide statistical uncertainty guarantees for runway detection in vision-based landing systems (VLS). Using fine-tuned YOLOv5 and YOLOv6 models on aerial imagery, we apply conformal prediction to quantify localization reliability under user-defined risk levels. We also introduce Conformal mean Average Precision (C-mAP), a novel metric aligning object detection performance with conformal guarantees. Our results show that conformal prediction can improve the reliability of runway detection by quantifying uncertainty in a statistically sound way, increasing safety on-board and paving the way for certification of ML system in the aerospace domain.