Both learning and inference tasks on Bayesian networks are NP-hard in general. Bounded tree-width Bayesian networks have recently received a lot of attention as a way to circumvent this complexity issue; however, while inference on bounded tree-width networks is tractable, the learning problem remains NP-hard even for tree-width~2. In this paper, we propose bounded vertex cover number Bayesian networks as an alternative to bounded tree-width networks. In particular, we show that both inference and learning can be done in polynomial time for any fixed vertex cover number bound $k$, in contrast to the general and bounded tree-width cases; on the other hand, we also show that learning problem is W[1]-hard in parameter $k$. Furthermore, we give an alternative way to learn bounded vertex cover number Bayesian networks using integer linear programming (ILP), and show this is feasible in practice.

Graph neural networks (GNNs) have shown superiority in many prediction tasks over graphs due to their impressive capability of capturing nonlinear relations in graph-structured data. However, for node classification tasks, often, only marginal improvement of GNNs has been observed in practice over their linear counterparts. Previous works provide very few understandings of this phenomenon. In this work, we resort to Bayesian learning to give an in-depth investigation of the functions of non-linearity in GNNs for node classification tasks. Given a graph generated from the statistical model CSBM, we observe that the max-a-posterior estimation of a node label given its own and neighbors' attributes consists of two types of non-linearity, the transformation of node attributes and a ReLU-activated feature aggregation from neighbors.

Our goal is to deploy a high-accuracy system starting with zero training examples. We consider an "on-the-job" setting, where as inputs arrive, we use real-time crowdsourcing to resolve uncertainty where needed and output our prediction when confident. As the model improves over time, the reliance on crowdsourcing queries decreases. We cast our setting as a stochastic game based on Bayesian decision theory, which allows us to balance latency, cost, and accuracy objectives in a principled way. Computing the optimal policy is intractable, so we develop an approximation based on Monte Carlo Tree Search. We tested our approach on three datasets-- named-entity recognition, sentiment classification, and image classification. On the NER task we obtained more than an order of magnitude reduction in cost compared to full human annotation, while boosting performance relative to the expert provided labels. We also achieve a 8% F1 improvement over having a single human label the whole set, and a 28% F1 improvement over online learning.

In this paper, we introduce a score-based diffusion model appr oach for adaptively learning the time-evolving solutions of stochastic partial differential equat ions (SPDEs) through recursive Bayesian inference. Partial differential equations (PDEs) are fundamental tools for modeling the dynamic behavior of complex physical systems. While they have been widely suc cessful in scientific and engineering applications, many practical scenarios involve inherent unc ertainties due to limited physical knowledge and environmental variability. For example, in climate and meteorological modeling, uncertainties in initial conditions, boundary data, and subgrid-scale ph ysical processes can significantly affect the accuracy of predictions governed by PDEs such as the Navier-Stokes or advection-diffusion equations. Similarly, in porous media flow problems, spatial het erogeneity and limited characterization of subsurface properties -- such as permeability or porosity -- i ntroduce substantial uncertainty into models governed by Darcy's law and related PDEs, making accu rate prediction particularly challenging. To capture these uncertainty effects and support rel iable predictive analysis, it is essential to incorporate SPDEs into mathematical modeling framework . The numerical solution of SPDEs has thus become a central focus of the uncertainty quantifica tion (UQ) community, where significant efforts have been dedicated to developing efficient solvers tha t can accurately characterize and propagate uncertainty in high-dimensional, nonlinear dynamica l systems (see, e.g., [1, 2, 13, 21, 36, 42, 53] and the reference therein). Despite advances in SPDE solvers capable of quantifying unc ertainty, significant challenges remain.

Machine learning (ML) systems are increasingly deployed in high-stakes domains where reliability is paramount. This thesis investigates how uncertainty estimation can enhance the safety and trustworthiness of ML, focusing on selective prediction -- where models abstain when confidence is low. We first show that a model's training trajectory contains rich uncertainty signals that can be exploited without altering its architecture or loss. By ensembling predictions from intermediate checkpoints, we propose a lightweight, post-hoc abstention method that works across tasks, avoids the cost of deep ensembles, and achieves state-of-the-art selective prediction performance. Crucially, this approach is fully compatible with differential privacy (DP), allowing us to study how privacy noise affects uncertainty quality. We find that while many methods degrade under DP, our trajectory-based approach remains robust, and we introduce a framework for isolating the privacy-uncertainty trade-off. Next, we then develop a finite-sample decomposition of the selective classification gap -- the deviation from the oracle accuracy-coverage curve -- identifying five interpretable error sources and clarifying which interventions can close the gap. This explains why calibration alone cannot fix ranking errors, motivating methods that improve uncertainty ordering. Finally, we show that uncertainty signals can be adversarially manipulated to hide errors or deny service while maintaining high accuracy, and we design defenses combining calibration audits with verifiable inference. Together, these contributions advance reliable ML by improving, evaluating, and safeguarding uncertainty estimation, enabling models that not only make accurate predictions -- but also know when to say "I do not know".

We propose a joint learning framework for Byzantine-resilient spectrum sensing and secure intelligent reflecting surface (IRS)--assisted opportunistic access under channel state information (CSI) uncertainty. The sensing stage performs logit-domain Bayesian updates with trimmed aggregation and attention-weighted consensus, and the base station (BS) fuses network beliefs with a conservative minimum rule, preserving detection accuracy under a bounded number of Byzantine users. Conditioned on the sensing outcome, we pose downlink design as sum mean-squared error (MSE) minimization under transmit-power and signal-leakage constraints and jointly optimize the BS precoder, IRS phase shifts, and user equalizers. With partial (or known) CSI, we develop an augmented-Lagrangian alternating algorithm with projected updates and provide provable sublinear convergence, with accelerated rates under mild local curvature. With unknown CSI, we perform constrained Bayesian optimization (BO) in a geometry-aware low-dimensional latent space using Gaussian process (GP) surrogates; we prove regret bounds for a constrained upper confidence bound (UCB) variant of the BO module, and demonstrate strong empirical performance of the implemented procedure. Simulations across diverse network conditions show higher detection probability at fixed false-alarm rate under adversarial attacks, large reductions in sum MSE for honest users, strong suppression of eavesdropper signal power, and fast convergence. The framework offers a practical path to secure opportunistic communication that adapts to CSI availability while coherently coordinating sensing and transmission through joint learning.

This paper presents an improved exponential tail bound for Beta distributions, refining a result in [15]. This improvement is achieved by interpreting their bound as a regular Kullback-Leibler (KL) divergence one, while introducing a specific perturbation $η$ that shifts the mean of the Beta distribution closer to zero within the KL bound. Our contribution is to show that a larger perturbation can be chosen, thereby tightening the bound. We then extend this result from the Beta distribution to Dirichlet distributions and Dirichlet processes (DPs).

This work presents a statistical mechanics characterization of neural networks, motivated by the replica symmetry breaking (RSB) phenomenon in spin glasses. A Hopfield-type spin glass model is constructed from a given feedforward neural network (FNN). Overlaps between simulated replica samples serve as a characteristic descriptor of the FNN. The connection between the spin-glass description and commonly studied properties of the FNN -- such as data fitting, capacity, generalization, and robustness -- has been investigated and empirically demonstrated. Unlike prior analytical studies that focus on model ensembles, this method provides a computable descriptor for individual network instances, which reveals nontrivial structural properties that are not captured by conventional metrics such as loss or accuracy. Preliminary results suggests its potential for practical applications such as model inspection, safety verification, and detection of hidden vulnerabilities.

Quantum magnetic sensing based on spin systems has emerged as a new paradigm for detecting ultra-weak magnetic fields with unprecedented sensitivity, revitalizing applications in navigation, geo-localization, biology, and beyond. At the heart of quantum magnetic sensing, from the protocol perspective, lies the design of optimal sensing parameters to manifest and then estimate the underlying signals of interest (SoI). Existing studies on this front mainly rely on adaptive algorithms based on black-box AI models or formula-driven principled searches. However, when the SoI spans a wide range and the quantum sensor has physical constraints, these methods may fail to converge efficiently or optimally, resulting in prolonged interrogation times and reduced sensing accuracy. In this work, we report the design of a new protocol using a two-stage optimization method. In the 1st Stage, a Bayesian neural network with a fixed set of sensing parameters is used to narrow the range of SoI. In the 2nd Stage, a federated reinforcement learning agent is designed to fine-tune the sensing parameters within a reduced search space. The proposed protocol is developed and evaluated in a challenging context of single-shot readout of an NV-center electron spin under a constrained total sensing time budget; and yet it achieves significant improvements in both accuracy and resource efficiency for wide-range D.C. magnetic field estimation compared to the state of the art.

In this work, we evaluate the potential of Large Language Models (LLMs) in building Bayesian Networks (BNs) by approximating domain expert priors. LLMs have demonstrated potential as factual knowledge bases; however, their capability to generate probabilistic knowledge about real-world events remains understudied. We explore utilizing the probabilistic knowledge inherent in LLMs to derive probability estimates for statements regarding events and their relationships within a BN. Using LLMs in this context allows for the parameterization of BNs, enabling probabilistic modeling within specific domains. Our experiments on eighty publicly available Bayesian Networks, from healthcare to finance, demonstrate that querying LLMs about the conditional probabilities of events provides meaningful results when compared to baselines, including random and uniform distributions, as well as approaches based on next-token generation probabilities. We explore how these LLM-derived distributions can serve as expert priors to refine distributions extracted from data, especially when data is scarce. Overall, this work introduces a promising strategy for automatically constructing Bayesian Networks by combining probabilistic knowledge extracted from LLMs with real-world data. Additionally, we establish the first comprehensive baseline for assessing LLM performance in extracting probabilistic knowledge.