Bayesian

Bayesian optimal experimental design (BOED) selects experiments to maximize information gain about model parameters. However, in decision-critical settings, reducing parameter uncertainty does not necessarily improve downstream decisions, as only specific parameter directions relevant to the objective truly matter. We propose GoBOED, a goal-driven BOED framework that directly optimizes experimental designs for a specified decision-making objective. GoBOED combines an amortized variational posterior surrogate with a differentiable convex decision layer, enabling gradient-based design optimization that is fully decision-focused. We theoretically show that GoBOED gradients are insensitive to parameter directions irrelevant to the decision objective, providing a formal justification for why goal-driven design achieves equivalent decision quality over a wider set of experimental designs than information-gain maximization. Empirically, across source localization, epidemic management, and pharmacokinetic control, GoBOED identifies designs that better align with downstream decision objectives and reveals that near-optimal design windows are substantially wider than those predicted by goal-agnostic BOED approaches.

Bayesian inverse design provides a principled framework for inferring aerodynamic geometries from sparse flow observations while quantifying uncertainty. However, its practical use in computational fluid dynamics (CFD) is severely limited by the cost of repeated high-fidelity simulations required for gradient-based Markov chain Monte Carlo (MCMC) sampling. While surrogate models are commonly proposed to reduce this cost, their effect on posterior geometry and uncertainty, especially for shock-dominated flows, remains poorly understood. In this work, we demonstrate that neural operator surrogates can be embedded directly within the MCMC inference loop while preserving posterior structure. Using a fully Bayesian inverse formulation of quasi-one-dimensional nozzle flow, we demonstrate that geometry parameterization plays a decisive role in identifiability and posterior conditioning, with cubic B-splines yielding stable and physically meaningful uncertainty estimates. Building on this formulation, a Deep Operator Network trained on CFD-generated data is substituted for the CFD solver within a No-U-Turn Sampler, while keeping the likelihood model, priors, and sampling configuration unchanged. Across sparse to fully observed regimes, surrogate-based inference reproduces the posterior geometry and uncertainty trends of the CFD reference. As a result of surrogate integration, total inference time is reduced to under one second, corresponding to a speedup exceeding three orders of magnitude. In addition, a direct inverse neural operator is examined as a deterministic alternative for inverse design, enabling single-shot geometry reconstruction without posterior sampling. These results demonstrate that neural operator-accelerated Bayesian inference enables practical, uncertainty-aware inverse design workflows for aerodynamic applications.

Evidential Deep Learning (EDL) has emerged as an efficient, sampling-free strategy for uncertainty estimation. A series of EDL variants have been proposed to address specific limitations of the original framework, achieving notable success. However, the underlying theoretical structure of EDL and the relationships among these variants have received limited systematic investigation. In this work, we establish a principled theoretical foundation for EDL by interpreting it within a generalized Bayesian framework that includes prior specification, posterior update, and training objective. We further characterize evidential uncertainty from a Bayesian distributional uncertainty viewpoint, established via asymptotic analysis. Building on this perspective, we further propose Generalized Evidential Deep Learning (GEDL), a unified and extensible framework that explicitly disentangles the roles of individual components and systematically relates GEDL to existing variants. Extensive experiments demonstrate that GEDL yields comparable results on classification, uncertainty estimation and OOD detections, with theoretical grounding.

Concerns about algorithmic bias and fairness have increased as artificial intelligence has been incorporated into high-stakes decision-making. Traditional Naive Bayes classifiers, while efficient and interpretable, lack fairness-awareness mechanisms and perpetuate historical biases in sensitive domains such as hiring, credit scoring, and criminal justice. This study develops a fairness-aware extension of the Naive Bayes classifier that mitigates bias while maintaining computational efficiency. We propose the Bias Mitigating Naive Bayes (BMNB) classifier, integrating in-processing and post-processing interventions. The in-processing stage employs a blended likelihood approach combining group-specific and pooled likelihood estimates through a tunable blending parameter alpha to balance fairness and accuracy. The post-processing stage applies output calibration with adaptive thresholding to fine-tune group-specific decision boundaries. Experimental results indicate that BMNB attains Disparate Impact (DI) values of 1.000, 1.171, and 0.997 and Equal Opportunity Difference (EOD) values of -0.217, -0.226, and -0.053 on the Adult, ProPublica, and Framingham datasets, respectively, while maintaining computational efficiency. Ablation studies confirm that the combination of blended likelihood and adaptive thresholding yields superior performance compared to either technique in isolation.

In online reinforcement learning, data scarcity creates epistemic uncertainty that makes robustness important early in learning, whereas sufficient exploration is needed to learn the true-environment optimal policy. We study this time-varying robustness--exploration trade-off through a quantile Bayesian risk-aware Markov decision process (BR-MDP), in which the quantile level controls how posterior uncertainty enters the Bellman backup. We characterize this control through an asymptotic normality result for the difference between the quantile BR-MDP value and the value in the true environment. The result implies that upper/lower-tail quantiles induce optimism/pessimism towards epistemic uncertainty, and the magnitude of the optimism/pessimism decreases as data accumulate. Building on this characterization, we propose an online Bayesian risk-aware algorithm with an adaptive quantile schedule that emphasizes robustness early and gradually encourages exploration of less-visited state--action pairs. We establish sublinear Bayesian regret bounds with respect to both the true optimal value and the optimal BR-MDP robust value. Numerical experiments demonstrate strong performance in both exploration-demanding and exploration-costly environments.

Conformal prediction (CP) gives distribution-free coverage for modern vision and language models, but it is often forced to make a ranking decision from a single unstable nonconformity score. Standard CP uses one realization, while average-then-calibrate variants smooth multiple realizations into a point estimate. Both options discard the inconsistency that can help identify whether a candidate is indeed stable. A weak answer can enter the conformal set even if the evidence is not strong, simply because one posterior sample or prompt phrasing made it look strong. But variability can help distinguish a stable signal from noise-driven fluctuations. We describe an empirical Bayes conformal prediction framework that uses $r$-values to convert score variability into an uncertainty informed nonconformity score. The resulting $r$-value estimates how likely a candidate's latent score belongs to the top-ranked group after accounting for both its mean score and its uncertainty. It admits both a closed-form Normal-Normal empirical Bayes estimator and a nonparametric posterior-sampling estimator. Using the $r$-value as the nonconformity score preserves the target conformal coverage while provably reducing the inclusion of high variance false candidates under mild regularity conditions. Across image classification, CLIP-based VLM benchmarks, and LLMs, we show that $r$-value conformal prediction preserves target coverage while improving ranking stability and reducing set size when variability is informative, and reverting to CP-like behavior when variability vanishes.

Multi-Agent Debate (MAD) improves LLM-agent accuracy but suffers from rapid context growth, limiting scalability in larger multi-agent settings. Existing methods prune low-utility communications using prior signals, such as token-level log-likelihoods or LLM self-reported confidence. However, these signals become unreliable under hallucination, degrading the accuracy of MAD methods that rely on them. We propose SVR-MAD, a Bayesian-inspired MAD framework that treats pre-debate signals as priors and debate outcomes as posterior-style evidence for estimating agent correctness. SVR-MAD uses this evidence to incrementally construct the communication graph, prioritizing agents whose answers survive peer challenges. Experiments across multiple LLMs and benchmarks show that SVR-MAD reduces token cost by up to 61% while matching or improving accuracy relative to the most accurate competing MAD baseline.

Symbolic regression with genetic programming (GPSR) may suffer from overfitting and structural bloat, especially when noise is present. In this paper we evaluate description length (DL) and fractional Bayes factor (FBF) criteria as principled, data-efficient alternatives to heuristics for selecting compact expressions that generalise well. We implement DL using a Fisher-information-based parameter encoding and compare it to AIC and BIC across multiple datasets, including noisy synthetic benchmarks and real-world regression problems. We study three search/selection strategies: (i) multi-objective search for accuracy and program length followed by DL/FBF selection; (ii) multi-objective search using DL directly as an objective; and (iii) single-objective optimisation with DL/FBF as the fitness. Across datasets we find that DL/FBF post-selection improves test performance compared to AIC/BIC baseline and that BIC in combination with the same function complexity penalty from DL/FBF produces similar results. In contrast, using DL/FBF directly as a fitness function in single-objective GPSR frequently induces premature convergence to overly simple models. We conclude with practical guidance for using DL/FBF as robust model-selection tools in genetic programming workflows.

Heterogeneous sensor fusion is vital for detecting, localizing, and classifying CBRNE threats. However, individual sensors are often only capable of detecting a subset of relevant threats with varying reliability or can even provide only indirect threat indications, making threat classification challenging. Furthermore, high clutter rates on the sensor side present a great challenge for fusion systems. Additionally, the limited availability of high quality datasets hinders the advancement of learning-based detection and classification models in smart sensors. To mitigate these sensor related shortcomings, a context-aware and domain knowledge-enhanced fusion process is proposed. First, a novel evidence hierarchy is established that enables modeling of direct, indicative, and contextual information. Second, contextual information about the environment is introduced into the fusion process, by collecting, processing, and exploiting OSINT inputs. Third, all levels of the evidence hierarchy are used to craft a Bayesian threat type classification mechanism with domain knowledge-informed priors. The proposed methodology is evaluated in simulated scenarios, and the results demonstrate the benefit of the proposed fusion approach in terms of robustness to clutter and prior mismatch, with an overall classification accuracy of up to 95%.

Magnetic resonance imaging (MRI) is highly susceptible to patient motion due to its relatively long acquisition times and the fact that data are acquired sequentially in k-space. Even small patient movements introduce phase inconsistencies across measurements, leading to severe artifacts such as blurring, ghosting, and geometric distortions that can compromise diagnostic quality. Retrospective motion compensation remains challenging, particularly in accelerated acquisitions, due to the ill-posed nature of the joint reconstruction and motion estimation problem. In this work, we propose a unified Bayesian framework for motion-compensated 3D MRI that jointly estimates the anatomical image, rigid-body motion parameters, and coil sensitivity maps directly from motion-corrupted k-space data. Our approach integrates pretrained 3D complex-valued score-based diffusion models as expressive anatomical image priors within a physics-based forward model. Inference is performed by alternating diffusion posterior image updates with efficient proximal optimization steps for motion and coil sensitivity estimation, enabling fully unsupervised reconstruction without the need for paired motion-free training data. Experiments on simulated and real-motion brain MRI datasets demonstrate that the proposed method achieves improved image quality and motion robustness compared to state-of-the-art classical and learning-based motion correction techniques, particularly in the presence of severe motion and high acceleration.

The high cost and data scarcity in scientific exploration have motivated the use of large language models (LLMs) as knowledge-driven components in Bayesian optimization (BO). However, existing approaches typically embed LLMs directly into the sampling or surrogate modeling pipeline, without fully leveraging their significantly lower evaluation cost compared to real-world experiments. To address this limitation, we propose LLM-Accelerated Bayesian Optimization (LABO), a framework that combines LLM predictions with experimental observations within a single BO loop. LABO employs a gating criterion to dynamically balance the reliance on LLM predictions versus actual experiments. By leveraging inexpensive LLM evaluations to broadly explore the search space and reserving costly real experiments only for regions with high uncertainty, LABO achieves more sample-efficient optimization. We provide a theoretical analysis with a cumulative regret bound that formalizes this efficiency gain. Empirical results across diverse scientific tasks demonstrate that LABO consistently outperforms existing methods under identical experimental budgets. Our results suggest that LABO offers a practical and theoretically grounded approach for integrating LLMs into scientific discovery workflows.

Reinforcement learning with verifiable rewards has become the standard recipe for improving LLM reasoning, but the dominant algorithm GRPO assigns a single trajectory-level advantage to every token, diluting the signal at pivotal reasoning steps and injecting noise at uninformative ones. Critic-free alternatives derived from on-policy distillation supply per-token signals through oracle-conditioned likelihood ratios, yet apply each signal in isolation from the trajectory-level evidence accumulated up to that position. We propose Oracle-Prompted Policy Optimization (OPPO), which rests on a single observation: the oracle signal used by prior distillation-style methods for local discrimination is also the natural Bayesian update of the model's belief about eventual success. Accumulating the signal along a trajectory yields, in closed form and at the cost of one extra forward pass, a running estimate of the success probability at every position, together with a token-level advantage that requires no learned value network and no additional rollouts. A first-order analysis factorizes the advantage into the per-token discrimination signal used by distillation methods modulated by a state weight that concentrates credit on genuinely pivotal tokens, with a directional variance-reduction guarantee. The framework admits two estimators differing only in which model scores the evidence: a \textit{self-oracle} that reuses the student and recovers the on-policy distillation reward as a strict special case, and a \textit{teacher-oracle} that delegates scoring to a stronger frozen model. On two base LLMs across seven mathematics, science, and code reasoning benchmarks, OPPO improves over GRPO, DAPO, and SDPO by up to $+6.0$ points on AMC'23 and $+5.2$ points on AIME'24, with gains that widen monotonically with response length.

State-space models (SSMs) are powerful probabilistic tools for modeling time-varying systems with latent dynamics. Inference in SSMs involves the estimation of latent states and parameters. In this work, we focus on parameter inference, which for SSMs is in general a very challenging problem due to the intractability of the likelihood. Recently, neural estimation methods, such as sequential neural likelihood (SNL), have shown promising results in Bayesian inference problems. In this paper, we show that SNL, when applied to the SSM setting, suffers important limitations, such as requiring a large amount of simulated samples to achieve a moderate performance, scaling poorly with sequence length, while not being amortized. We then introduce a novel inference algorithm called truncated-SNL (T-SNL), which addresses the limitations of SNL. Our algorithm is more accurate, more stable and robust during training, more scalable to longer temporal sequences, and can be amortized when new observations become available. Our experiments show that T-SNL is sample-efficient, robust, and flexible algorithm which outperforms other approaches.

Test-time adaptation (TTA) methods improve model performance under distribution shift but lack formal guarantees connecting shift magnitude to prediction reliability. We develop a PAC-Bayesian framework yielding generalization bounds explicitly parameterized by the maximum mean discrepancy (MMD) between source and target distributions. Our principal contribution is interpreting MMD-balls around the source distribution as credal sets in Walley's imprecise probability theory, yielding natural epistemic uncertainty quantification. We establish: (i) a PAC-Bayesian bound with an MMD-dependent shift penalty under an RKHS-Lipschitz loss assumption; (ii) a finite-sample version via MMD concentration; (iii) a uniform worst-case risk bound over all distributions in the credal set, with a lower-upper risk decomposition; and (iv) geodesic preservation bounds explaining why kernel-guided adaptation protects local feature geometry. The credal set interpretation separates epistemic from aleatoric uncertainty and provides a principled decision criterion for when adaptation is warranted.

In this work, we extend the Equilibrium Propagation framework to skew-gradient systems and show an equivalence between deep Energy-Based Models and Hamiltonian neural networks. We focus on networks of diffusively coupled Fitzhugh-Nagumo neurons as a prototypical example. We show that since stationary solutions of the Fitzhugh-Nagumo model are described by self-adjoint operators, the methods of equilibrium propagation for performing credit assignment can be applied. Furthermore, for Fitzhugh-Nagumo networks with the topology of a deep residual network, we show that the steady state solutions admit a (spatial) Hamiltonian, and thus the methods of Hamiltonian Echo Backpropagation can be applied. We end by deriving an explicit layer-wise Hamiltonian recurrence relation governing inference for stationary solutions of both deep Fitzhugh-Nagumo networks and deep Energy-Based Models.

Compositional score-based approaches to simulation-based inference (SBI) approximate the posterior over a shared parameter given $n$ independent observations by aggregating individually learned posterior scores: currently, there are two main propositions of such methods (Geffner et al. (2023), Linhart et al. (2026)). As the resulting composite score does not correspond to the score of any distribution along the forward diffusion path of the true multi-observation posterior, sampling from it via a reverse SDE leads to an irreducible bias. Annealed Langevin dynamics provides a principled alternative: it treats the composite score as the genuine score of a sequence of tractable bridging densities and samples from them in succession. When properly tuned, it could lead to a controllable bias. However, its hyperparameters, namely step sizes, the number of steps per level, and the number of annealing levels, have so far been chosen empirically. We derive Wasserstein bounds for annealed Langevin with approximate scores and translate them into explicit decision rules for these hyperparameters that guarantee a prescribed sampling accuracy, while highlighting different theoretical aspects of each composite score formulation. In the Gaussian setting, we obtain closed-form expressions for all relevant quantities and prove that the bridging densities of Linhart et al. (2026) consistently admit larger step sizes and require fewer total Langevin steps than those of Geffner et al. (2023). Furthermore, we show empirically that the tuning obtained in the Gaussian setting generalizes to more complex problems, thus providing a well-understood and theoretically grounded starting point for practitioners using compositional score-based approaches.

We study transformers' generalization behavior on boolean domains from the perspective of the Fourier spectra of their target functions. In contrast to prior work (Edelman et al., 2022; Trauger & Tosh, 2024), which derived generalization bounds from Rademacher complexity, we investigate the feasibility of obtaining generalization bounds via PAC-Bayes theory. We show that sparse spectra concentrated on low-degree components enable low-sharpness constructions with good generalization properties. Our idea is to show the existence of flat minima implementing any boolean function of sparsity no greater than the context length, and then apply a PAC-Bayes bound to an idealized low-sharpness learner, resulting in a non-vacuous generalization bound. We use this to give a formal account of why chain-of-thought improves generalization for high-degree target functions, and show that the complexity parameters in our bound can be efficiently estimated via property testing. We evaluate predictions empirically and conduct a mechanistic interpretability study to support the realism of our theoretical construction in real transformers.

Learning from implicit feedback in recommender systems is fundamentally challenged by pervasive label noise. While conventional denoising approaches often discard noisy instances to ensure robustness, this strategy inevitably suffers from low data utilization. Alternative methods that employ a Bayes-label transition matrix (BLTM) can leverage all available data, but their estimates tend to be biased in practical recommendation scenarios. To address these limitations, this paper proposes a Robust GMM-weighted Bayes-label Transition Matrix framework (RGBT). Our solution utilizes a Gaussian Mixture Model (GMM) to derive instance-specific reliability scores, which systematically calibrate the BLTM estimation to mitigate bias. Theoretical analysis confirms that our approach, by leveraging the BLTM framework with GMM calibration, simultaneously ensures full sample utilization, delivers consistent estimation, and critically, achieves a significant reduction in estimation variance. Extensive experiments on multiple real-world and synthetically flipped datasets demonstrate that RGBT not only utilizes noisy samples more effectively than mainstream reliable sample-based denoising methods, but also achieves significantly superior calibration capability of the transition matrix compared to state-of-the-art transition matrix-based denoising approaches.

Latent Gaussian models (LGMs) are a popular class of Bayesian hierarchical models that include Gaussian processes, as well as certain spatial models and mixed-effect models. Efficient Bayesian inference of LGMs often requires marginalizing out the latent variables. For LGMs with a non-Gaussian likelihood, exact marginalization is not possible and a popular approach is to do approximate marginalization with an integrated Laplace approximation (ILA). Using ILA produces an approximate posterior which, in some settings, can differ significantly from the correct posterior, which impacts downstream applications. We propose an importance sampling scheme to correct the error introduced by ILA. By increasing the number of samples in importance sampling, the posterior with ILA converges to the correct posterior. This idea is realized with various techniques, including pseudo-marginalization, quasi-Monte Carlo and randomized quasi-Monte Carlo. We implement our methods in an automatic differentiation framework to support gradient-based algorithms when doing inference on the hyperparameters. For the latter, we specifically consider the use of Hamiltonian Monte Carlo. We demonstrate the benefits of reduced error in various applied models.

Backpropagation with gradient descent is a common optimization strategy employed by most neural network architectures in machine learning. However, finding optimal hyperparameters to guide training has proven challenging. While it is widely acknowledged that selecting appropriate parameters is crucial for avoiding overfitting and achieving unbiased outcomes, this choice remains largely based on empirical experiments and experience. This paper presents a new probabilistic framework for the learning rate, a key parameter in stochastic gradient descent. The framework develops classic Bayesian statistics into a double-Bayesian decision mechanism involving two antagonistic Bayesian processes. A theoretically optimal learning rate can be derived from these two processes and used for stochastic gradient descent. Experiments across various classification, segmentation, and detection tasks corroborate the practical significance of the theoretically derived learning rate. The paper also discusses the ramifications of the proposed double-Bayesian framework for network training and model performance.

Medical image segmentation faces a fundamental challenge in continual learning: data arrives sequentially from heterogeneous sources, yet effective continual learning requires discovering which tasks share sufficient structure to benefit from joint learning. Existing methods either apply uniform constraints across all tasks, causing catastrophic forgetting when tasks conflict, or require predefined task groupings that cannot anticipate future task diversity. We introduce MedCRP-CL, a framework that performs online task structure discovery and structure-aware continual learning. Leveraging the Chinese Restaurant Process (CRP), our method dynamically infers task groupings from clinical text prompts as tasks arrive, without requiring predefined cluster counts or access to future tasks. We term these discovered groupings semantic modalities, as they capture finer-grained structure than physical imaging modalities by integrating anatomical region and pathological context. Guided by this discovered structure, we maintain semantic modality-specific LoRA adapters regularized by intra-modality EWC, ensuring parameter isolation across dissimilar task groups while facilitating knowledge transfer within similar ones. The framework is also replay-free, storing only aggregate statistics rather than raw patient data. Experiments on 16 medical segmentation tasks across four imaging modalities demonstrate that MedCRP-CL achieves 73.3% Dice score with only 4.1% forgetting, outperforming the best baseline by 8.0% while requiring 6$\times$ fewer parameters. Code is available at https://github.com/zygao930/MedCRP-CL.

Coding and computation remain major bottlenecks in Markov chain Monte Carlo (MCMC) workflows, especially as modern sampling algorithms have become increasingly complex and existing probabilistic programming systems remain limited in model support, extensibility, and composability. We introduce \textbf{AI4BayesCode}, an extensible LLM-driven system that translates natural-language Bayesian model descriptions into runnable, validated MCMC samplers. To improve reliability, AI4BayesCode adopts a modular design that decomposes models into modular sampling blocks and maps each block to a built-in sampling component, reducing the need to implement complex sampling algorithms from scratch. Reliability is further improved through pre-generation validation of model specifications and post-generation validation of generated sampler code. AI4BayesCode also introduces a novel recursively stateful coding paradigm for MCMC, allowing modular sampling components, potentially developed by different contributors, to be composed coherently within larger MCMC procedures. We develop a benchmark suite to evaluate AI4BayesCode for sampler-generation. Experiments show that AI4BayesCode can implement a wide range of Bayesian models from natural-language descriptions alone. As an open-ended system, its capability can continue to expand with improvements in the underlying AI agent and the addition of new built-in blocks.

Bayesian latent space models offer a principled approach to network representation, but rely on correct specification of both geometry and link function. Real-world networks often violate these assumptions, exhibiting geometric mismatch and structural anomalies that break standard metric properties. We show that such misspecification pushes the data-generating distribution outside the model class, causing Bayesian inference to become overconfident and poorly calibrated. To address this, we propose a generalized posterior framework for random geometric graphs. We introduce Link-Sequential R-SafeBayes, a method that exploits dyadic conditional independence to estimate prequential risk and adaptively tune posterior regularization. Experiments on synthetic and real-world networks demonstrate improved calibration, better link prediction performance, and a reliable criterion for selecting latent geometries across Euclidean, spherical, and hyperbolic spaces.

A unified framework for learning under covariate shift is presented, in which a constrained density-ratio network approximates the Radon-Nikodym derivative $r^\star = dP/dQ$ and feeds an anytime PAC-Bayes generalization certificate. A change-of-measure identity decomposes the gap between target risk and importance-weighted source risk into a ratio-bias term governed by $\|r_θ- r^\star\|_{L^2(Q)}$ and a generalization-gap term governed by the variability of the weighted loss. Normalization and moment-matching identities are enforced as hard integral constraints through an augmented-Lagrangian scheme, with a second-moment penalty controlling the effective sample size. PAC-Bayes is instantiated on the weighted risk in a fixed-time regime that yields Bernoulli-KL bounds, identifies the network-weighted Gibbs posterior as the unique KL-regularized minimizer, and quantifies stability under $L^2(Q)$ perturbations of the learned ratio, and is then strengthened by geometric peeling to an anytime certificate uniform in $t \geq t_{\min}$. A pre-registered two-campaign protocol combining a patch test against analytic ground truth with a real-data deployment validates the framework: the network produces calibrated ratios, reduces target $0/1$ loss against unweighted ERM and classical direct ratio-estimation baselines, and attains the anytime certificate. A single fixed-time coverage failure is recorded, with per-split coverage aligning one-to-one with the magnitude of the label shift, confirming that the covariate-only assumption is operationally tight rather than a defect of the certificate.

Dempster's rule is a fundamental tool for combining belief functions from distinct and reliable sources. However, its intersection-based semantics imposes strong structural restrictions, which limits its flexibility in handling complex source states and diverse information fusion scenarios. To overcome this limitation, we propose a reversible transformation, derived from the isopignistic principle, between belief functions and a possibilistic structure defined on the power set. In this transformation, the relationships among subsets are explicitly characterized by a belief evolution network, which provides a more flexible representation of evidential information beyond the conventional mass function structure. On this basis, we further introduce the triangular norm family to develop a general and adaptive evidential information fusion framework. Unlike fusion methods rooted in Dempster semantics, the proposed framework supports more flexible combination behaviors and exhibits advantages in non-distinct source fusion, conflict management, parametric combination design, and heterogeneous information fusion.

A "path-based sensor" produces a single observation along a continuous path. For example, a boolean path-based sensor returns a single "1" if an event of interest is detected at any point along the path and a "0" otherwise. Notably, a "1" provides no direct information about where along the path the event(s) may have occurred. Previous work has demonstrated that observations from multiple path-based sensors can be fused to create a Bayesian belief map over the spatial locations of the underlying event or phenomenon. Moreover, path planning can employ Shannon information theory to accelerate the rate of convergence of the belief map. In this paper, we present a new method to update the belief map based on a path-based sensor observation, and then plan paths to increase information gain. In contrast to prior work that approximates the posterior by averaging over the alternative event histories, we introduce a Bayesian Network (BN) formulation that models the probabilistic relationships between the latent variables and path-based sensor measurements, enabling a more principled Bayesian belief update. We consider static hazard detection in a communication-denied environment as a representative problem setting. The event of a robot returning from its path corresponds to a path-based hazard sensor reading of "0" (hazard not detected), while a robot failing to return corresponds to a reading of "1" (hazard detected). We consider false positives and false negatives. We find that the new method leads to quicker convergence of the belief map than prior work in both single- and multi-robot cases.

Many consumer decisions are repeated choices under uncertainty. Standard models capture these decisions using Bayesian learning and dynamic programming: consumers update beliefs from feedback and use those beliefs to guide future choices. In many markets, however, learning does not restart when consumers enter a new context: prior experience with a brand, product, or provider can shape beliefs in later, related decisions. We study this cross-context knowledge transfer, or meta-learning, in sequential choice. We design a hierarchical laboratory task in which participants repeatedly choose among airlines across routes and observe noisy binary outcomes. Reduced-form evidence shows that participants improve not only within routes, but also across routes: they choose better airlines earlier in later routes and reduce pseudo-regret. To identify the mechanism behind this transfer, we compare human choices to a no-transfer benchmark and a fully integrated Bayesian meta-learning benchmark. In particular, we introduce a class of boundedly rational meta dynamic programming policies, BRMDP(D), that approximate full integration using a limited number of hyper-posterior draws, denoted by D. Trial-by-trial likelihood comparisons show that low-D boundedly rational meta-learning, especially BRMDP(1), fits participant behavior better than both no transfer and fully integrated Bayesian transfer. Consumers, therefore, transfer brand-level regularities across contexts, but through coarse representations of prior uncertainty. The findings imply that models of consumer learning should allow for approximate cross-context transfer, and that managerial counterfactuals based on either no-transfer or fully integrated learning can be misleading.

Survival analysis provides a powerful statistical framework for modeling time-to-event outcomes in the presence of censoring. However, selecting an appropriate estimator from the many specialized survival approaches often requires substantial methodological and domain expertise. We introduce SurvivalPFN, a prior-data fitted network that amortizes Bayesian inference for censored observations through in-context learning. SurvivalPFN is pretrained on a diverse family of synthetic, identifiable, and right-censored data-generating processes, enabling it to amortize survival analysis in a single forward pass during inference. As a result, the model adapts to the effective complexity of each dataset without task-specific training or hyperparameter tuning, avoids restrictive parametric assumptions, and produces calibrated survival distributions. In a large-scale benchmark spanning 61 datasets, 21 methods, and 5 evaluation metrics, SurvivalPFN achieves strong predictive performance and often improves upon established survival models. These results suggest that SurvivalPFN offers a principled and practical foundation model for survival analysis, with potential applications in high-impact domains such as healthcare, finance, and engineering (https://github.com/rgklab/SurvivalPFN).

We consider amortized Bayesian inference for nonlinear inverse problems in settings where only samples from the joint distribution of parameters and observations are available. Classical methods such as Markov chain Monte Carlo require solving a new inference problem for each observation, which can be computationally prohibitive when inference must be repeated many times. We propose a transport-based approach that learns an observation-dependent map pushing forward a reference measure to approximate the posterior distribution. The map is trained by minimizing an averaged energy-distance objective between the true posterior and the learned pushforward. This formulation is likelihood-free, requiring only joint samples, and avoids density evaluation, invertibility constraints, and Jacobian determinant computations. For function-space inverse problems with Gaussian priors, we parameterize the transport map as the identity plus a perturbation in the Cameron-Martin space of the prior, preserving absolute continuity with respect to the prior. In infinite-dimensional settings, the map is represented using neural operators. We illustrate the method on a finite-dimensional nonlinear inverse problem and two PDE-constrained inverse problems arising in porous medium flow and seismic inversion. The results show that the learned transport captures posterior structure, including multimodality and dominant modes, while enabling fast posterior sampling for new observations.

bde is a user-friendly Python package for Bayesian Deep Ensembles with a particular focus on tabular data. Built on an efficient JAX implementation of the sampling-based inference method Microcanonical Langevin Ensembles (MILE), it provides scikit-learn compatible estimators for fast training, efficient Markov Chain Monte Carlo sampling, and uncertainty quantification in both regression and classification tasks.

Integration against a probability distribution given its unnormalized density is a central task in Bayesian inference and other fields. We introduce new methods for approximating such expectations with a small set of weighted samples -- i.e., a quadrature rule -- constructed via an interacting particle system that minimizes maximum mean discrepancy (MMD) to the target distribution. These methods extend the classical mean shift algorithm, as well as recent algorithms for optimal quantization of empirical distributions, to the case of continuous distributions. Crucially, our approach creates dynamics for MMD minimization that are invariant to the unknown normalizing constant; they also admit both gradient-free and gradient-informed implementations. The resulting mean shift interacting particle systems converge quickly, capture anisotropy and multi-modality, avoid mode collapse, and scale to high dimensions. We demonstrate their performance on a wide range of benchmark sampling problems, including multi-modal mixtures, Bayesian hierarchical models, PDE-constrained inverse problems, and beyond.

This work studies lung tumor growth prediction from sparse and irregular longitudinal computed tomography (CT) observations with measurement variability. A Bayesian physics-informed neural network is developed by combining Gompertz growth dynamics with low-dimensional Bayesian inference in the log-volume domain. The framework employs a two-stage inference strategy combining maximum a posteriori (MAP) estimation and Hamiltonian Monte Carlo (HMC) sampling to estimate posterior predictive distributions and uncertainty intervals. The method was evaluated on longitudinal data from the National Lung Screening Trial (30 patients). Results show that the model captures heterogeneous tumor growth patterns while maintaining reasonable prediction accuracy under limited observations. Compared with deterministic modeling approaches, the proposed approach additionally provides calibrated uncertainty estimates. The inferred posterior parameter correlations were consistent with expected biological growth behavior. The proposed framework achieved a cohort-level log-space RMSE of approximately 0.20 together with well-calibrated 95% credible interval coverage across 30 patients. These findings suggest that Bayesian physics-informed modeling may be useful for uncertainty-aware tumor growth assessment when only limited longitudinal follow-up scans are available.

Synapses are densely packed submicron structures that dynamically reorganize during learning and memory formation. Longitudinal \textit{in vivo} imaging of fluorescently tagged synaptic receptors offers a promising opportunity to study large-scale synaptic dynamics and how these processes are disrupted in neurological disease. However, in vivo imaging with 2-photon microscopy uses low laser power and therefore suffers from low signal-to-noise ratio (SNR) and high shot noise, nonlinear tissue motion between days, nonstationary fluctuations in synaptic fluorescence, and significant blur induced by the microscope point spread function (PSF). Together, these factors make it challenging to detect and track synapses, especially in regions with high synaptic density. This paper presents a novel template-based framework for modeling synapses as varying luminance point sources that move under a nonlinear tissue deformation. Taking a unified Bayesian approach, we apply this model to microscopy data by deriving a posterior that incorporates a diffeomorphic mapping for domain warping, a Gaussian point spread function for the imaging process, and a Poisson observation model for raw photon counts. The Bayesian solution simultaneously: (1) Constructs a probabilistic template of synapse locations, (2) denoises and deconvolves the image data, (3) infers fluorescence intensities, (4) performs diffeomorphic image registration to correct for tissue motion, and (5) provides confidence regions for these parameter estimates. We demonstrate the framework on both a 2D+t simulated dataset and a 3D+t longitudinal \textit{in vivo} microscopy dataset of fluorescent synapses imaged in a mouse over two weeks.

Deep neural networks generalize well despite being heavily overparameterized, in apparent contradiction with classical learning theory based on uniform convergence over fixed hypothesis spaces. Uniform bounds over the entire parameter space are vacuous in this regime, and recent work has shown that non-vacuous guarantees can be recovered by restricting attention to the part of parameter space that the algorithm actually visits. This survey paper organizes this line of work around three steps: extending PAC-Bayesian theory to random, data-dependent hypothesis sets (arXiv:2404.17442); refining the complexity term with geometric and topological descriptors of the optimization trajectory, including fractal dimensions, alpha-weighted lifetime sums, and positive magnitude (arXiv:2006.09313, arXiv:2302.02766, arXiv:2407.08723); and replacing the resulting information-theoretic terms by stability assumptions (arXiv:2507.06775). We unify these contributions around a single template inequality and a head-to-head comparison of the resulting bounds.

Modern Bayesian optimization and adaptive sampling methods increasingly rely on nonlinear parametric models, yet theoretical guarantees for such models under adaptive data collection remain limited. Existing analyses largely focus on Gaussian processes, kernel machines, linear models, or linearized neural approximations, leaving a gap between theory and the nonlinear models used in practice. We develop a kernel based framework for analyzing regularized nonlinear parametric models trained on adaptively collected data. Our approach uses kernels over the parameter space to induce reproducing kernel Hilbert space structures over the corresponding model class, yielding confidence bounds for models trained with broad classes of regularized convex losses. We show how these bounds can support convergence guarantees for nonlinear acquisition and surrogate models, including randomized regularized policies that select points by maximizing a trained random model. These results provide a unified route to analyzing nonlinear parametric models in Bayesian optimization and related adaptive optimization settings.

Model merging aims to combine multiple task-specific expert models into a single model without joint retraining, offering a practical alternative to multi-task learning when data access or computational budget is limited. Existing methods, however, face two key limitations: (1) they overlook the valuable inductive bias of strong anchor models and estimate the merged weights from scratch, and (2) they rely on a shared hyperparameter setting across different modules of the network, lacking a global optimization strategy. This paper introduces Bayesian Model Merging (BMM), a plug-and-play bi-level optimization framework, where the inner level formulates the model merging as an activation-based Bayesian regression under a strong prior induced by an anchor model, yielding an efficient closed-form solution; and the outer level leverages a Bayesian optimization procedure to search module-specific hyperparameters globally based on a small validation set. Furthermore, we reveal a key alignment between activation statistics and task vectors, enabling us to derive a data-free variant of BMM that estimates the Gram matrix for regression without any auxiliary data. Across extensive benchmarks, including up to 20-task merging in vision and 5-task merging in language, BMM consistently outperforms all plug-and-play anchor baselines (e.g., TA, WUDI-Merging, and TSV). In particular, on the ViT-L/14 benchmark for 8-task merging, a single merged model reaches 95.1, closely matching the average performance of eight task-specific experts (95.8).

Approximate Bayesian inference typically revolves around computing the posterior parameter distribution. In practice, however, the main object of interest is often a model's predictions rather than its parameters. In this work, we propose to bypass the parameter posterior and focus directly on approximating the posterior predictive distribution. We achieve this by drawing inspiration from self-training within self-supervised and semi-supervised learning. Essentially, we quantify a Bayesian model's predictive uncertainty by refitting on self-predicted data. The idea is strikingly simple: If a model assigns high likelihood to self-predicted data, these predictions are of low uncertainty, and vice versa. This yields a deterministic, sampling-free approximation of the posterior predictive. The modular structure of our Self-Supervised Laplace Approximation (SSLA) further allows us to plug in different prior specifications, enabling classical Bayesian sensitivity (w.r.t. prior choice) analysis. In order to bypass expensive refitting, we further introduce an approximate version of SSLA, called ASSLA. We study (A)SSLA both theoretically and empirically in regression models ranging from Bayesian linear models to Bayesian neural networks. Across a wide array of regression tasks with simulated and real-world datasets, our methods outperform classical Laplace approximations in predictive calibration while remaining computationally efficient.

For stochastic process models, parameter inference is often severely bottlenecked by computationally expensive likelihood functions. Simulation-based inference (SBI) bypasses this restriction by constructing amortized surrogate likelihoods, but most SBI methods assume a black-box data generating process. While these surrogates are exact in the limit of infinite training data, practical scenarios force a strict tradeoff between model quality and simulation cost. In this work, we loosen the black-box assumption of SBI to improve this tradeoff for structured stochastic process models. Specifically, for neural network likelihood surrogates trained via probabilistic classification, we propose to augment the standard binary cross-entropy loss with exact score information $\nabla_θ\log p(x \mid θ)$ and adaptive weighting based on loss gradients. We evaluate our approach on case studies involving network dynamics and spatial processes, demonstrating that our method improves surrogate quality at a drastically lower computational cost than generating more training data. Notably, in some cases, our approach achieves downstream inference performance equivalent to a 10x increase in training data with less than a 1.1x increase in training time.

Constant Liar (CL), Kriging Believer (KB), and fantasy models are widely used for batch selection in parallel Bayesian Optimization, yet a unified theory explaining their effectiveness and conditions under which they fail has been lacking. We identify efficient conditioning as the key surrogate property the ability to update predictions in closed form when data is augmented. We prove that Gaussian Processes satisfy this requirement, producing provably distinct batch points with separation of order l, and that this holds for any acquisition function monotonically non decreasing in posterior uncertainty (EI, UCB, PI), with qualitatively similar behavior for Thompson Sampling. We unify CL, KB, and fantasy models as instances of a single conditioning mechanism differing only in the lie value distribution, and draw quantitative connections to Local Penalization (LP) and qualitative connections to Determinantal Point Processes (DPPs). To disentangle model structure from optimizer randomness, we introduce the Structural Diversity Diagnostic (SDD), a reusable methodology for testing surrogate compatibility. Experiments on Hartmann6D, Ackley 8D, Levy10D, and SVM hyperparameter tuning validate all theoretical predictions: CL or KBs implicit penalty matches or outperforms explicit LP greedy conditioning achieves convergence on par with joint qEI efficient conditioning extends to Multiquadric RBF networks; and parametric surrogates produce degenerate batches even when fully retrained (random forests), while neural networks regain diversity only at 15x the wall clock cost of GP conditioning. Robustness is confirmed across multiple initial datasets and under observation noise.

Human-in-the-loop Bayesian optimization (HITL BO) methods utilize human expertise to improve the sample-efficiency of BO. Most HITL BO methods assume that a domain expert can quantify their knowledge, for instance by pinpointing query locations or specifying their prior beliefs about the location of the maximum as a probability distribution. However, since human expertise is often tacit and cannot be explicitly quantified, we consider a setting where domain knowledge of an expert is elicited via pairwise comparisons of designs. We interpret the expert's pairwise judgements as noisy evidence about the values of the observable objective function and develop a principled method for combining the information obtained via direct observations and pairwise queries. Specifically, we derive a cost-aware value-of-information acquisition function that balances direct observations against pairwise queries. The proposed method approaches the convex hull of the trajectories of the individual information sources: when pairwise queries are cheap it substantially improves sample-efficiency over observation-only BO, and when pairwise queries are costly or noisy, it recovers the performance of standard BO by relying on direct observations alone.

Collaborative machine learning involves training high-quality models using datasets from a number of sources. To incentivize sources to share data, existing data valuation methods fairly reward each source based on its data submitted as is. However, as these methods do not verify nor incentivize data truthfulness, the sources can manipulate their data (e.g., by submitting duplicated or noisy data) to artificially increase their valuations and rewards or prevent others from benefiting. This paper presents the first mechanism that provably ensures (F) collaborative fairness and incentivizes (T) truthfulness at equilibrium for Bayesian models. Our mechanism combines semivalues (e.g., Shapley value), which ensure fairness, and a truthful data valuation function (DVF) based on a validation set that is unknown to the sources. As semivalues are influenced by others' data, we introduce an additional condition to prove that a source can maximize its expected data values in coalitions and semivalues by submitting a dataset that captures its true knowledge. Additionally, we discuss the implications and suitable relaxations of (F) and (T) when the mediator has a limited budget for rewards or lacks a validation set. Our theoretical findings are validated on synthetic and real-world datasets.

Bayesian optimization (BO) is a widely used framework for optimizing expensive black-box functions, commonly based on Gaussian process (GP) surrogate models. Its effectiveness relies on uncertainty quantification that is both sharp (informative) and well-calibrated along the BO trajectory. In practice, GP kernel hyperparameters are unknown and are refit online from sequentially collected (non-i.i.d.) data, which can yield miscalibrated or overly conservative uncertainty and lies outside the fixed-kernel assumptions of standard BO regret theory. We propose Online Sharp-Calibrated Bayesian Optimization (OSCBO), a BO algorithm that adaptively balances GP sharpness and calibration by casting hyperparameter selection as a constrained online-learning problem. We also show that OSCBO preserves sublinear regret bounds by leveraging the theoretical guarantees of the underlying online learning algorithm. Empirically, OSCBO performs competitively across synthetic and real-world benchmarks, ranking among the strongest methods in final simple regret while maintaining robust cumulative-regret behavior.

We study the information-theoretic limits of learning a one-hidden-layer teacher network with hierarchical features from noisy queries, in the context of knowledge transfer to a smaller student model. We work in the high-dimensional regime where the teacher width $k$ scales linearly with the input dimension $d$ -- a setting that captures large-but-finite-width networks and has only recently become analytically tractable. Using a heuristic leave-one-out decoupling argument, validated numerically throughout, we derive asymptotically sharp characterizations of the Bayes-optimal generalization error and individual feature overlaps via a system of closed fixed-point equations. These equations reveal that feature learnability is governed by a sequence of sharp phase transitions: as data grows, teacher features become recoverable sequentially, each through a discontinuous jump in overlap. This sequential acquisition underlies a precise notion of \textit{effective width} $k_c$ -- the number of learnable features at a given data budget $n$ -- which unifies two distinct scaling regimes: a feature-learning regime in which the Bayes-optimal generalization error $\varepsilon^{\rm BO}$ scales as $n^{1/(2β)-1}$, and a refinement regime in which it scales as $n^{-1}$, where $β>1/2$ is the exponent of the power-law feature hierarchy. Both laws collapse to the single relation $\varepsilon^{\rm BO}=Θ(k_c d/n)$. We further show empirically that a student trained with \textsc{Adam} near the effective width $k_c$ achieves these optimal scaling laws (up to a small algorithmic gap), and provide an information-theoretic account of the associated scaling in model size.

Reliable inference requires that artificial intelligence (AI) models provide trustworthy uncertainty estimates, not merely accurate predictions. Recent advances in Bayesian learning have made significant progress toward this goal, and growing concerns about computational overhead have jointly shifted the design criterion from reliability alone to the co-design of reliability and efficiency, i.e., reducing computational overhead while preserving trustworthy uncertainty quantification. This thesis develops a unified framework from two perspectives to address the central question: can we efficiently perform reliable inference?

A central challenge in large-scale decision-making under incomplete information is estimating reliable probabilities. Recent approaches use Large Language Models (LLMs) to generate explanatory factors and coarse-grained probability estimates, which are then refined by a Naïve Bayes model over factor combinations. However, sparse factor spaces often yield unknown'' predictions, while expanding factors increases noise and spurious correlations, weakening conditional independence and degrading reliability. To address these limitations, we propose \textsc{Anchor}, an aggregated Bayesian inference framework over a hierarchical factor space. It constructs dense factor hierarchies through iterative generation and clustering, maps contexts via hierarchical retrieval and refinement, and augments Naïve Bayes with a Causal Bayesian Network to model latent factor dependencies. Experiments show that \textsc{Anchor} markedly reduces unknown'' predictions and produces more reliable probability estimates than direct LLM baselines, achieving state-of-the-art performance while significantly reducing time and token overhead.

Frontier coding agents read configuration files (CLAUDE$.$md, AGENTS$.$md, Cursor Rules) at session start and are expected to follow the conventions inside them. Practitioners assume that structural choices (file size, instruction position, file architecture, contradictions in adjacent files) measurably affect adherence. We report a systematic factorial study of these choices using four manipulated variables, measuring compliance with a trivial target annotation across 1,650 Claude Code CLI sessions (16,050 function-level observations) on two TypeScript codebases, three frontier models (primarily Sonnet 4.6, with Opus 4.6 as a CLI-matched cross-model check and Opus 4.7 reported descriptively under a CLI-version confound), and five coding tasks. We use mixed-effects models with a Bayesian companion. None of the four structural variables or three two-way interactions produces a detectable contrast after multiple-testing correction. Size and conflict nulls are supported by affirmative-null Bayes factors (BF10 between 0.05 and 0.10); position and architecture nulls are failures to reject without Bayes-factor support. The largest effect we measured is within-session: each additional function the agent generates is associated with approximately 5.6% lower odds of compliance per step (OR = 0.944) within the session-length range we tested, though the relationship is non-monotonic rather than a constant per-step effect. This reproduces on a second TypeScript codebase and on Opus 4.6 at matched configuration; it was identified during analysis rather than pre-specified. Within the conditions tested, file-structure variables did not produce detectable contrasts; compliance varies systematically between coding tasks and across each session's sequence of generated functions.

Modern predictive systems encode beliefs that can act as useful prior information for statistical inference in data-limited settings. Using them for prior construction introduces a tradeoff: an informative prior built from a predictive model can sharpen inference from limited data, but also risks propagating error from the model into the posterior. We propose a framework for AI-informed prior elicitation that mitigates this tension by rectifying the AI-induced law that generates synthetic data before using it to inform a prior. The rectified law can be embedded into synthetic data-driven prior elicitation techniques, including as a base measure in a Dirichlet process (DP) prior on the data-generating process. We refer to the resulting prior and corresponding posterior as the rectified AI prior and rectified AI posterior. We establish Gaussian asymptotics for the rectified AI posterior under non-vanishing prior strength and derive a first-order expression for its centering bias. Our rectified AI priors substantially reduce bias compared to standard approaches, improve the coverage of credible intervals, and make AI-powered prior information more reliable. We additionally apply the rectified AI prior to a real skin disease classification task and show that it can meaningfully boost predictive performance.

Diffusion language models (DLMs) have recently emerged as a promising alternative to autoregressive models, primarily due to their ability to enable parallel decoding. Despite this advantage, most existing DLMs rely on a fixed generation length specified prior to decoding, which restricts their flexibility in real-world applications. While a few recent works attempt to support flexible-length generation, they typically suffer from notable limitations: some require costly retraining to accommodate variable-length outputs, while others depend solely on local confidence signals during decoding. Such local criteria fail to capture the evolving structure of the sequence, often resulting in suboptimal generation quality. In this paper, we propose a training-free, Bayesian structured decoding framework that formulates flexible-length generation as a dynamic structural inference problem. Our approach formulates flexible-length generation as a dynamic structural inference problem, jointly computing the expansion length, the block boundaries, and the decoding schedule. At each window expansion step, the method integrates local uncertainty with structural signals via a unified mechanism that supports dynamic structured generation, including both flexible block expansion and block organization, while maintaining coherence. Extensive experiments across multiple benchmarks demonstrate that our approach significantly improves generation quality and flexibility over existing fixed-length and flexible-length baselines. These results highlight the advantage of Bayesian structured decoding for diffusion language model, providing a principled and efficient solution for structured text generation.

Bayesian optimization (BO) is an efficient framework for optimizing expensive black-box functions. However, it is typically formulated as learning an end-to-end mapping from inputs to scalar objectives, thereby discarding the potentially rich information whenever a structured system output is available. In this work, we study Bayesian optimization over a vector-valued operator with structured measurements, where each measurement observes multidimensional or functional outputs, e.g., trajectories or spatial fields, rather than a single scalar value. The objective is then defined as a linear functional of these measurements. This allows each observation to reveal substantially richer information about the underlying system compared to scalar observations. Assuming the unknown operator lies in a vector-valued reproducing kernel Hilbert space (RKHS), we derive high-probability concentration bounds for the kernel ridge regression (KRR) estimator directly in the measurement space, characterizing uncertainty in a general Hilbert space. Building on these results, we propose an algorithm based on the upper confidence bound (UCB) acquisition function with regret guarantees under mild assumptions, recovering sublinear rates for common kernels. Empirically, we demonstrate that leveraging structured measurements leads to improved sample efficiency by enabling efficient transfer of information across objectives and adaptation to time-varying settings.

The problem of predicting unobserved entries in a binary matrix, known as 1-bit matrix completion, has found diverse applications in fields such as recommendation systems. In this study, we develop an empirical Bayes method for 1-bit matrix completion motivated by the Efron--Morris estimator, a matrix generalization of the James--Stein estimator that shrinks singular values toward zero. The proposed method exploits the underlying low-rank structure of binary matrices, drawing parallels with multidimensional item response theory. Simulation studies and real-data applications demonstrate that the proposed method achieves a superior balance of predictive accuracy, calibration reliability (uncertainty quantification), and computational efficiency compared to existing methods.

In this (work in progress) paper, we present Bounded Pragmatic Listener (or BPL), a cognitively grounded Bayesian framework for modelling susceptibility to information disorder. BPL extends Rational Speech Act theory with three cognitively motivated bounds derived from the bounded rationality literature with a) a recursion depth bound (that emphasises working memory limits);b) a prior compression parameter (which is oriented at capturing information bottleneck); and c) an availability sample size (that operationalises importance sampling with saliency-weighted proposals). This allows us to test predictions about misinformation susceptibility, annotator disagreement, and the differential vulnerability to mis-, dis-, and mal-information as defined in the Information Disorder framework. We validate BPL on the LIAR and MultiFC benchmarks showcasing competitive veracity classification and experimental support for the depth-mismatch paradox.

Bayesian neural networks are typically trained against the evidence lower bound (ELBO), whose Jensen gap closes only when the variational posterior is exact. We instead train by local consistency: gradient descent on the Bethe free energy, driving the belief at every factor toward agreement with its neighbours rather than placing a loss on the output. The resulting objective scores each observation by its own predictive density: a strictly proper rule whose optimum is the true conditional, for any likelihood with a tractable predictive convolution. Instantiated with a Gaussian last layer over a deterministic backbone, exact inference appears as one known corner: the neural-linear marginal likelihood. That corner is evidence-optimal; the shared-cavity, free-routed interior is predictive-optimal, improving NLL and calibration over it. This instance, SCROLL (Shared-Cavity fRee-rOuting Last-Layer), is a single-pass Bayesian neural network: batchable, any-likelihood, and implicitly empirical-Bayes-prior precision, observation noise, covariance, and backbone fit in one gradient pass. Prior work enters the interior only through the ELBO and its Jensen gap, even in this conjugate setting. At a single training run and forward pass per architecture-where the validation-tuned conventional references cross-validate $λ$ and ensembles pay $5$-$50\times$ at inference-a fixed SCROLL variant is best-or-tied on NLL and calibration on 7/8 UCI regression benchmarks, and best on 4/5 across three large tabular datasets (up to 515k examples) and two frozen text/vision embeddings.

Human-AI teams play a pivotal role in improving overall system performance when neither the human nor the model can achieve such performance on their own. With the advent of powerful and accessible Generative AI models, several mundane tasks have morphed into Human-AI team tasks. From writing essays to developing advanced algorithms, humans have found that using AI assistance has led to an accelerated work pace like never before. In classification tasks, where the final output is a single hard label, it is crucial to address the combination of human and model output. Prior work elegantly solves this problem using Bayes rule, using the assumption that human and model output are conditionally independent given the ground truth. Specifically, it discusses a combination method to combine a single deterministic labeler (the human) and a probabilistic labeler (the classifier model) using the model's instance-level and the human's class-level calibrated probabilities.

Causal inference, especially in observational studies, relies on untestable assumptions about the true data-generating process. Sensitivity analysis helps us determine how robust our conclusions are when we alter these underlying assumptions. Existing frameworks for sensitivity analysis are concerned with worst-case changes in assumptions. In this work, we argue that using such pessimistic criteria can often become uninformative or lead to conclusions contradicting our prior knowledge about the world. To demonstrate this claim, we generalize the recent s-value framework (Gupta & Rothenhäusler, 2023) to estimate the sensitivity of three different common assumptions in causal inference. Empirically, we find that, indeed, worst-case conclusions about sensitivity can rely on unrealistic changes in the data-generating process. To overcome this, we extend the s-value framework with a new sensitivity analysis criterion: Bayesian Sensitivity Value (BSV), which computes the expected sensitivity of an estimate to assumption violations under priors constructed from real-world evidence. We use Monte Carlo approximations to estimate this quantity and illustrate its applicability in an observational study on the effect of diabetes treatments on weight loss.

These notes introduce the theory of susceptibilities as developed in [arXiv:2504.18274, arXiv:2601.12703] for interpreting neural networks. The susceptibility of an observable $φ$ to a data perturbation is defined as a derivative of a posterior expectation, which by the fluctuation--dissipation theorem equals a posterior covariance. Different choices of $φ$ yield different objects: per-sample losses give the influence matrix (the Bayesian influence function of [arXiv:2509.26544]), while component-localized observables give the structural susceptibility matrix that pairs model components with data patterns. The susceptibility matrix is (up to a factor of $nβ$) the Jacobian of the map from data distributions to structural coordinates; its pseudo-inverse provides a linearized solution to the patterning problem of [arXiv:2601.13548]: finding data perturbations that produce a desired structural change. We motivate the theory from its statistical-mechanical foundations, then give a detailed exposition of susceptibilities, their empirical estimators, and their connection to the geometry of the loss landscape.

Confidence sequences based on test martingales provide time-uniform uncertainty quantification for the mean of bounded IID observations without parametric distributional assumptions. Their practical efficiency, however, depends strongly on the choice of martingale updates, and many existing constructions do not exploit prior information about plausible data-generating distributions or mean values. We propose a Bayes-assisted framework that uses a Bayesian working predictive model to adaptively construct confidence sequences. For each candidate mean and time point, the predictive distribution selects, among valid one-step martingale factors, the update maximising predictive expected log-growth; validity is therefore preserved even when the prior or working model is misspecified. We prove that if the predictive distribution is Wasserstein-consistent, the resulting procedure is asymptotically log-optimal, matching the per-sample log-growth of an oracle procedure with access to the true distribution. We instantiate the framework using robust predictives based on Dirichlet-process mixtures and Bayesian exponentially tilted empirical likelihood. Experiments on synthetic data, sequential best-arm identification for LLM evaluation, and prediction-powered inference show that informative priors can substantially reduce confidence-sequence width and sampling effort while retaining anytime-valid coverage.

In this work, we study TabPFN as a training-free, modular summary network for simulation-based Bayesian inference (SBI). Tabular foundation models such as TabPFN are pretrained on broad families of synthetic tabular data-generating processes and adapt at test time through in-context learning, making them natural candidates for SBI, where posterior estimation often depends on learning informative summaries of simulated observations. We propose PFN-NPE: a general recipe that uses a pretrained TabPFN encoder as a fixed summary network for simulator outputs, then pairs the resulting summaries with a downstream inference head chosen for the problem. With normalizing flows as the default inference head, PFN-NPE matches established posterior approximation methods and sometimes outperforms them. More importantly, diagnostic probes show that the TabPFN-derived summaries often preserve useful posterior location and marginal information. These analyses also reveal a limitation in that TabPFN-derived summaries may struggle to represent the joint posterior structure even when the marginals are well recovered. Still, our experiments show that TabPFN can serve as an effective summary network across a diverse set of SBI settings, with the inference network left modular and task-dependent.

When applying Bayesian optimization (BO) to scientific workflow, a major yet often overlooked source of uncertainty is the task itself -- namely, what to optimize and how to evaluate it -- which can evolve as evidence accumulates. We introduce Generate-Select-Refine (GSR), a open-ended BO framework that alternates between task generation and task optimization. Starting from a user-provided seed task, GSR generates new tasks in a coarse-to-fine manner while a task-acquisition function schedules optimization. Asymptotically, it concentrates evaluations on the best task, incurring only logarithmic regret overhead relative to single-task BO. We apply GSR to new product development, chemical synthesis scaling, algorithm analysis, and patent repurposing, where it outperforms existing LLM-based optimizers.

Bounded fitting is an attractive paradigm for learning logical formulas from labeled data examples that offers PAC-style generalization guarantees and can often be implemented leveraging SAT solvers. It has been successfully applied to learning concepts of the description logic ALC. We study bounded fitting for learning concepts in expressive description logics that extend ALC with inverse roles, qualified number restrictions, and feature comparisons. We investigate under which conditions bounded fitting keeps its favorable theoretical properties in this setting, and implement it using a SAT solver. We compare our tool with state-of-the-art concept learners with encouraging results, demonstrating that it is a practical approach to expressive concept learning.

Physics-informed neural networks (PINNs) provide a mesh-free framework for solving PDE-constrained inverse problems, but their extension to Bayesian inversion still faces a fundamental difficulty: prior distributions are typically defined in the weight space of neural networks, whereas physically meaningful prior assumptions are more naturally expressed in function space. In this study, we introduce a unified framework, termed functional-prior-based approaches to Bayesian PDE-constrained inversion using physics-informed neural networks (fpBPINN), to incorporate functional priors into Bayesian PINN-based inversion. We consider two complementary approaches. The first is a functional-prior-informed Bayesian PINN (FPI-BPINN), in which a neural network weight prior is learned to be consistent with a prescribed functional prior, and Bayesian inference is subsequently performed in weight space. The second is function-space particle-based variational inference for PINNs (fParVI-PINN), which performs Bayesian estimation using ParVI directly in function space. We also show that random Fourier features (RFF) play an important role in representing Gaussian functional priors with neural networks and in improving posterior approximation. We applied the proposed approaches to one-dimensional seismic traveltime tomography and two-dimensional Darcy-flow permeability inversion. These numerical experiments showed that both approaches accurately estimated posterior distributions, highlighting the significance of introducing physically interpretable functional priors into Bayesian PINN-based inverse problems. We also identified the contrasting advantages of FPI-BPINN and fParVI-PINN, namely flexibility and accuracy, respectively.