Bayesian

Foundation models of brain activity promise a new frontier for in silico neuroscience by emulating neural responses to complex stimuli across tasks and modalities. A natural next step is to ask whether these models can also be used in reverse. Can we recover a stimulus or its properties from synthetic brain activity? We study this question in a proof-of-concept setting using TRIBEv2. We pair the brain emulator with large language models (LLMs) that generate news headlines from linguistic parameters such as valence, arousal, and dominance. We then use simulation-based inference to learn a probabilistic mapping from brain maps to latent stimulus parameters. Our results show that these parameters can be recovered from predicted brain maps, validating the quality of neural encodings. They also show that LLMs can serve as controllable stimulus generators for simulated experiments. Together, these findings provide a step toward decoding and inverse design with foundation brain models.

Expressway video anomaly detection is essential for safety management. However, identifying anomalies across diverse scenes remains challenging, particularly for far-field targets exhibiting subtle abnormal vehicle motions. While Vision-Language Models (VLMs) demonstrate strong semantic reasoning capabilities, processing global frames causes attention dilution for these far-field objects and incurs prohibitive computational costs. To address these issues, we propose VIBES, an asynchronous collaborative framework utilizing VLMs guided by Bayesian inference. Specifically, to overcome poor generalization across varying expressway environments, we introduce an online Bayesian inference module. This module continuously evaluates vehicle trajectories to dynamically update the probabilistic boundaries of normal driving behaviors, serving as an asynchronous trigger to precisely localize anomalies in space and time. Instead of processing the continuous video stream, the VLM processes only the localized visual regions indicated by the trigger. This targeted visual input prevents attention dilution and enables accurate semantic reasoning. Extensive evaluations demonstrate that VIBES improves detection accuracy for far-field anomalies and reduces computational overhead, achieving high real-time efficiency and explainability while demonstrating generalization across diverse expressway conditions.

The health condition of components in civil infrastructures can be described by various discrete states according to their performance degradation. Inferring these states from measurable responses is typically an ill-posed inverse problem. Although Bayesian methods are well-suited to tackle such problems, computing the posterior probability density function (PDF) presents challenges. The likelihood function cannot be analytically formulated due to the unclear relationship between discrete states and structural responses, and the high-dimensional state parameters resulting from numerous components severely complicates the computation of the marginal likelihood function. To address these challenges, this study proposes a novel Bayesian inversion paradigm for discrete variables based on Probabilistic Graphical Models (PGMs). The Markov networks are employed as modeling tools, with model parameters learned from data and structural topology prior. It has been proved that inferring this PGM produces the same probabilistic estimation as the posterior PDF derived from Bayesian inference, which effectively solves the above challenges. The inference is accomplished by Graph Neural Networks (GNNs), and a graph property-based GNN training strategy is developed to enable accurate inference across varying graph scales, thereby significantly reducing the computational overhead in high-dimensional problems. Both synthetic and experimental data are used to validate the proposed framework

The proliferation of agentic artificial intelligence has outpaced the conceptual tools needed to characterize agency in computational systems. Prevailing definitions mainly rely on autonomy and goal-directedness. Here, we argue for a minimal notion open to principled inspection given three criteria: intentionality as action grounded in beliefs and desires, rationality as normatively coherent action entailed by a world model, and explainability as action causally traceable to internal states; we subsequently instantiate these as a partially observable Markov decision process under a variational framework wherein posterior beliefs, prior preferences, and the minimization of expected free energy jointly constitute an agentic action chain. Using a canonical T-maze paradigm, we evidence how empowerment, formulated as the channel capacity between actions and anticipated observations, serves as an operational metric that distinguishes zero-, intermediate-, and high-agency phenotypes through structural manipulations of the generative model. We conclude by arguing that as agents engage in epistemic foraging to resolve ambiguity, the governance controls that remain effective must shift systematically from external constraints to the internal modulation of prior preferences, offering a principled, variational bridge from computational phenotyping to AI governance strategy

Standard evaluations of Bayesian deep learning methods assume that metric estimates are reliable, but we show this assumption fails under data scarcity. Method rankings are not only unreliable at small $n$, but also dataset-dependent in ways that point estimates cannot reveal: the same method comparison yields $P(\mathrm{MCD} \prec \mathrm{Ensemble}) = 1.000$ at $n = 50$ on one dataset and remains below $0.95$ even at $n = 500$ on another. Across the datasets we consider, no universal sample size threshold exists, which is precisely why dataset-specific posterior inference is necessary. To address this, we use a Bayesian hierarchical model with method-specific variances to treat evaluation metrics as random variables across data realizations, and we use a predictive Minimum Detectable Difference curve to assess whether an observed gap would be detectable at a given training size. Across six Bayesian deep learning methods and five regression datasets, our results show that uncertainty-aware evaluation is necessary in low-data settings, because current evidence for method superiority and predictive detectability at the same training size can diverge substantially. Our framework provides practitioners with principled tools to determine whether their evaluation data is sufficient before drawing conclusions about method superiority.

Bayesian Optimal Experimental Design (BOED) provides a rigorous framework for decision-making tasks in which data acquisition is often the critical bottleneck, especially in resource-constrained settings. Traditionally, BOED typically selects designs by maximizing expected information gain (EIG), commonly defined through the Kullback-Leibler (KL) divergence. However, classical evaluation of EIG often involves challenging nested expectations, and even advanced variational methods leave the underlying log-density-ratio objective unchanged. As a result, support mismatch, tail underestimation, and rare-event sensitivity remain intrinsic concerns for KL-based BOED. To address these fundamental bottlenecks, we introduce an IPM-based BOED framework that replaces density-based divergences with integral probability metrics (IPMs), including the Wasserstein distance, Maximum Mean Discrepancy, and Energy Distance, resulting in a highly flexible plug-and-play BOED framework. We establish theoretical guarantees showing that IPM-based utilities provide stronger geometry-aware stability under surrogate-model error and prior misspecification than classical EIG-based utilities. We also validate the proposed framework empirically, demonstrating that IPM-based designs yield highly concentrated credible sets. Furthermore, by extending the same sample-based BOED template in a plug-and-play manner to geometry-aware discrepancies beyond the IPM class, illustrated by a neural optimal transport estimator, we achieve accurate optimal designs in high-dimensional settings where conventional nested Monte Carlo estimators and advanced variational methods fail.

Multivariate conformal prediction requires nonconformity scores that compress residual vectors into scalars while preserving certain implicit geometric structure of the residual distribution. We introduce a Multivariate Kernel Score (MKS) that produces prediction regions that explicitly adapt to this geometry. We show that the proposed score resembles the Gaussian process posterior variance, unifying Bayesian uncertainty quantification with the coverage guarantees of frequentist-type. Moreover, the MKS can be decomposed into an anisotropic Maximum Mean Discrepancy (MMD) that interpolates between kernel density estimation and covariance-weighted distance. We prove finite-sample coverage guarantees and establish convergence rates that depend on the effective rank of the kernel-based covariance operator rather than the ambient dimension, enabling dimension-free adaptation. On regression tasks, the MKS reduces the volume of prediction region significantly, compared to ellipsoidal baselines while maintaining nominal coverage, with larger gains at higher dimensions and tighter coverage levels.

Autonomous robots deployed in mass casualty incidents (MCI) face the challenge of making critical decisions based on incomplete and noisy perceptual data. We present an autonomous robotic system for casualty assessment that fuses outputs from multiple vision-based algorithms, estimating signs of severe hemorrhage, visible trauma, or physical alertness, into a coherent triage assessment. At the core of our system is a Bayesian network, constructed from expert-defined rules, which enables probabilistic reasoning about a casualty's condition even with missing or conflicting sensory inputs. The system, evaluated during the DARPA Triage Challenge (DTC) in realistic MCI scenarios involving 11 and 9 casualties, demonstrated a nearly three-fold improvement in physiological assessment accuracy (from 15% to 42% and 19% to 46%) compared to a vision-only baseline. More importantly, overall triage accuracy increased from 14% to 53%, while the diagnostic coverage of the system expanded from 31% to 95% of cases. These results demonstrate that integrating expert-guided probabilistic reasoning with advanced vision-based sensing can significantly enhance the reliability and decision-making capabilities of autonomous systems in critical real-world applications.

While diffusion priors generate high-quality posterior samples across many inverse problems, they are often trained on limited training sets or purely simulated data, thus inheriting the errors and biases of these underlying sources. Current approaches to finetuning diffusion models rely on a large number of observations with varying forward operators, which can be difficult to collect for many applications, and thus lead to overfitting when the measurement set is small. We propose a method for tuning a prior from only a single observation by combining existing diffusion priors into a single product-of-experts prior and identifying the exponents that maximize the Bayesian evidence. We validate our method on real-world inverse problems, including black hole imaging, where the true prior is unknown a priori, and image deblurring with text-conditioned priors. We find that the evidence is often maximized by priors that extend beyond those trained on a single dataset. By generalizing the prior through exponent weighting, our approach enables posterior sampling from both tempered and combined diffusion models, yielding more flexible priors that improve the trustworthiness of the resulting posterior image distribution.

Accurate condition monitoring of industrial equipment requires inferring latent degradation parameters from indirect sensor measurements under uncertainty. While traditional Bayesian methods like Markov Chain Monte Carlo (MCMC) provide rigorous uncertainty quantification, their heavy computational bottlenecks render them impractical for real-time process control. To overcome this limitation, we propose an AI-driven framework utilizing Simulation-Based Inference (SBI) powered by amortized neural posterior estimation to diagnose complex failure modes in heat exchangers. By training neural density estimators on a simulated dataset, our approach learns a direct, likelihood-free mapping from thermal-fluid observations to the full posterior distribution of degradation parameters. We benchmark this framework against an MCMC baseline across various synthetic fouling and leakage scenarios, including challenging low-probability, sparse-event failures. The results show that SBI achieves comparable diagnostic accuracy and reliable uncertainty quantification, while accelerating inference time by a factor of82$\times$ compared to traditional sampling. The amortized nature of the neural network enables near-instantaneous inference, establishing SBI as a highly scalable, real-time alternative for probabilistic fault diagnosis and digital twin realization in complex engineering systems.

The cost of simulator evaluations is a key practical bottleneck for Simulation Based Inference (SBI). In hierarchical settings with shared global parameters and exchangeable site-level parameters and observations, this structure can be exploited to improve simulation efficiency. Existing hierarchical SBI approaches factorise the posterior yet still simulate across multiple sites per training sample; We instead explore likelihood factorisation (LF) to train from single-site simulations. In LF sampling we learn a per-site neural surrogate of the simulator and then assemble synthetic multi-site observations to amortise inference for the full hierarchical posterior. Building on this, we propose Tokenised Flow Matching for Posterior Estimation (TFMPE), a tokenised flow matching approach that supports function-valued observations through likelihood factorisation. To enable systematic evaluation, we introduce a benchmark for hierarchical SBI. We validate TFMPE on this benchmark and on realistic infectious disease and computational fluid dynamics models, finding well-calibrated posteriors while reducing computational cost.

Digital twins of natural systems must remain aligned with physical systems that evolve over time, are only partially observed, and are typically modeled by mechanistic simulators whose parameters cannot be measured directly. In such settings, model adaptation is naturally posed as a simulation-based inference problem. However, sparse and indirect observations often fail to identify a unique and optimal calibration, leaving several simulator parameterizations compatible with the available evidence. This article presents a GFlowNet-based approach to model adaptation for digital twins of natural systems. We formulate adaptation as a generative modeling problem over complete simulator configurations, so that plausible parameterizations can be sampled with probability proportional to a reward derived from agreement between simulated and observed behavior. Using a controlled environment agriculture case study based on a mechanistic tomato model, we show that the learned policy recovers dominant regions of the adaptation landscape, retrieves strong calibration hypotheses, and preserves multiple plausible configurations under uncertainty.

Mixture-of-experts models provide a flexible framework for learning complex probabilistic input-output relationships by combining multiple expert models through an input-dependent gating mechanism. These models have become increasingly prominent in modern machine learning, yet their theoretical properties in the Bayesian framework remain largely unexplored. In this paper, we study Bayesian mixture-of-experts models, focusing on the ubiquitous softmax-based gating mechanism. Specifically, we investigate the asymptotic behavior of the posterior distribution for three fundamental statistical tasks: density estimation, parameter estimation, and model selection. First, we establish posterior contraction rates for density estimation, both in the regimes with a fixed, known number of experts and with a random learnable number of experts. We then analyze parameter estimation and derive convergence guarantees based on tailored Voronoi-type losses, which account for the complex identifiability structure of mixture-of-experts models. Finally, we propose and analyze two complementary strategies for selecting the number of experts. Taken together, these results provide one of the first systematic theoretical analyses of Bayesian mixture-of-experts models with softmax gating, and yield several theory-grounded insights for practical model design.

Large language models (LLMs) are increasingly used for conversational clinical decision support, yet they conflate next token prediction with probabilistic decision making. We argue that this conflation reflects an architectural limitation: such systems lack explicit posterior tracking, controllable abstention thresholds, and auditable reasoning chains. We introduce MoBayes, a Modular Bayesian dialogue framework that separates reasoning from language. The LLM acts only as a language interface, parsing patient conversation into structured observations, while a Bayesian module performs probabilistic inference over these observations to update posteriors, select follow-up questions via expected-information-gain and determine when to stop or defer through calibrated decision thresholds. This design enables explicit posterior tracking, controllable selective decision-making, and replaceable population-specific statistical backends without retraining the language model. Across empirical and LLM-generated knowledge bases, MoBayes outperforms standalone frontier LLM doctors, including matched model-family comparisons where inexpensive sensor models paired with MoBayes exceed larger autonomous models at lower cost. The advantage persists under adversarial patient communication styles and across varying diagnostic scenarios. These results suggest that reliable conversational clinical decision support systems should separate probabilistic reasoning from language generation rather than scaling model size alone. Code is available at https://anonymous.4open.science/r/MoBayes/

Complex-Valued Neural Networks (CVNNs) have significant advantages in handling tasks that involve complex numbers. However, existing CVNNs are unable to quantify predictive uncertainty. We propose, for the first time, dropout-based Bayesian Complex-Valued Neural Networks (BayesCVNNs) to enable uncertainty quantification for complex-valued applications, exhibiting broad applicability and efficiency for hardware implementation due to modularity. Furthermore, as the dual-part nature of complex values significantly broadens the design space and enables novel configurations based on layer-mixing and part-mixing, we introduce an automated search approach to effectively identify optimal configurations for both real and imaginary components. To facilitate deployment, we present a framework that generates customized FPGA-based accelerators for BayesCVNNs, leveraging a set of optimized building blocks. Experiments demonstrate the best configuration can be effectively found via the automated search, attaining higher performance with lower hardware costs compared with manually crafted models. The optimized accelerators achieve approximately 4.5x and 13x speedups on different models with less than 10% power consumption compared to GPU implementations, and outperform existing work in both algorithm and hardware aspects. Our code is publicly available at: https://github.com/zehuanzhang/BayesCVNN.git.

One of the most challenging problems in graph machine learning is generalizing across graphs with diverse properties. Graph neural networks (GNNs) face a fundamental limitation: they require separate training for each new graph, preventing universal generalization across diverse graph datasets. A critical challenge facing GNNs lies in their reliance on labeled training data for each individual graph, a requirement that hinders the capacity for universal node classification due to the heterogeneity inherent in graphs -- differences in homophily levels, community structures, and feature distributions across datasets. Inspired by the success of large language models (LLMs) that achieve in-context learning through massive-scale pre-training on diverse datasets, we introduce NodePFN. This universal node classification method generalizes to arbitrary graphs without graph-specific training. NodePFN learns posterior predictive distributions (PPDs) by training only on thousands of synthetic graphs generated from carefully designed priors. Our synthetic graph generation covers real-world graphs through the use of random networks with controllable homophily levels and structural causal models for complex feature-label relationships. We develop a dual-branch architecture combining context-query attention mechanisms with local message passing to enable graph-aware in-context learning. Extensive evaluation on 23 benchmarks demonstrates that a single pre-trained NodePFN achieves 71.27 average accuracy. These results validate that universal graph learning patterns can be effectively learned from synthetic priors, establishing a new paradigm for generalization in node classification.

We study the deterministic global optimization of trained Gaussian process posterior mean functions over hyperrectangular domains. Although the posterior mean function has a compact closed-form representation, its global optimization is challenging because it remains nonlinear and nonconvex. Existing exact deterministic approaches become increasingly difficult to scale as the number of training data points grows, leading to approximation-based methods that improve tractability by optimizing a modified (inexact) objective. In this work, we propose PALM-Mean, a piecewise-analytic lower-bounding framework embedded in reduced-space spatial branch-and-bound. At each node, kernel terms that are locally important are replaced by a sign-aware piecewise-linear relaxation in an appropriate scalar distance variable, while the remaining terms are bounded analytically in closed form. We show this hybrid approach yields a valid lower bound for the posterior mean, while limiting the size of the branch-and-bound subproblems. We establish validity of the node lower bounds and $\varepsilon$-global convergence of the resulting algorithm. Computational results on synthetic benchmarks and real-world application problems show that PALM-Mean improves scalability relative to representative general-purpose deterministic global solvers, particularly as the number of training data points increases.

Discovering optimal designs through sequential data collection is essential in many real-world applications. While Bayesian Optimization (BO) has achieved remarkable success in this setting, growing attention has recently turned to context-specific optimal design, formalized as Contextual Bayesian Optimization (CBO). Unlike BO, CBO is inherently more challenging as it must approximate an entire mapping from the context space to its corresponding optimal design, requiring simultaneous exploration across contexts and exploitation within each. In many modern applications, such tasks arise across multiple potentially heterogeneous but related clients, where collaboration can significantly improve learning efficiency. We propose CCBO, Collaborative Contextual Bayesian Optimization, a unified framework enabling multiple clients to jointly perform CBO with controllable contexts, supporting both online collaboration and offline initialization from peers' historical beliefs, with an optional privacy-preserving communication mechanism. We establish sublinear regret guarantees and demonstrate, through extensive simulations and a real-world hot rolling application, that CCBO achieves substantial improvements over existing approaches even under client heterogeneity. The code to reproduce the results can be found at https://github.com/cchihyu/Collaborative-Contextual-Bayesian-Optimization

We present the Bayesian Linguistic Forecaster (BLF), an agentic system for binary forecasting that achieves state-of-the-art performance on the ForecastBench benchmark. The system is built on three ideas. (1) Linguistic belief state: a semi-structured representation combining numerical probability estimates with natural-language evidence summaries, updated by the LLM at each step of an iterative tool-use loop. This contrasts with the common approach of appending all retrieved evidence to an ever-growing, unstructured context. (2) Hierarchical multi-trial aggregation: running $K$ independent trials and combining them using logit-space averaging shrinkage with a data-dependent prior. (3) Hierarchical calibration: Platt scaling with a hierarchical prior, which avoids over-shrinking extreme predictions for sources with skewed base rates. On 400 questions from the ForecastBench leaderboard, BLF outperforms all the top public methods, including Cassi, GPT-5, Grok~4.20, and Foresight-32B. Careful ablation studies, using mixed effects analysis to control for question variability (which accounts for 62% of the variance in performance), reveals that all 3 components contribute to the overall gains, but some components matter more than others, depending on the base LLM, and the setting (e.g.\ with or without a crowd prior). All our experiments are based on a robust back-testing framework which we develop, which has a leakage rate below 1.5%, and may be of independent interest.

Selection bias arises when the probability that an observation enters a dataset depends on variables related to the quantities of interest, leading to systematic distortions in estimation and uncertainty quantification. For example, in epidemiological or survey settings, individuals with certain outcomes may be more likely to be included, resulting in biased prevalence estimates with potentially substantial downstream impact. Classical corrections, such as inverse-probability weighting or explicit likelihood-based models of the selection process, rely on tractable likelihoods, which limits their applicability in complex stochastic models with latent dynamics or high-dimensional structure. Simulation-based inference enables Bayesian analysis without tractable likelihoods but typically assumes missingness at random and thus fails when selection depends on unobserved outcomes or covariates. Here, we develop a bias-aware simulation-based inference framework that explicitly incorporates selection into neural posterior estimation. By embedding the selection mechanism directly into the generative simulator, the approach enables amortized Bayesian inference without requiring tractable likelihoods. This recasting of selection bias as part of the simulation process allows us to both obtain debiased estimates and explicitly test for the presence of bias. The framework integrates diagnostics to detect discrepancies between simulated and observed data and to assess posterior calibration. The method recovers well-calibrated posterior distributions across three statistical applications with diverse selection mechanisms, including settings in which likelihood-based approaches yield biased estimates. These results recast the correction of selection bias as a simulation problem and establish simulation-based inference as a practical and testable strategy for parameter estimation under selection bias.

We derive explicit non-asymptotic PAC-Bayes generalization bounds for Gibbs posteriors, that is, data-dependent distributions over model parameters obtained by exponentially tilting a prior with the empirical risk. Unlike classical worst-case complexity bounds based on uniform laws of large numbers, which require explicit control of the model space in terms of metric entropy (integrals), our analysis yields posterior-averaged risk bounds that can be applied to overparameterized models and adapt to the data structure and the intrinsic model complexity. The bound involves a marginal-type integral over the parameter space, which we analyze using tools from singular learning theory to obtain explicit and practically meaningful characterizations of the posterior risk. Applications to low-rank matrix completion and ReLU neural network regression and classification show that the resulting bounds are analytically tractable and substantially tighter than classical complexity-based bounds. Our results highlight the potential of PAC-Bayes analysis for precise finite-sample generalization guarantees in modern overparameterized and singular models.

Physics-informed neural networks (PINNs) have emerged as a promising framework for solving inverse problems governed by partial differential equations (PDEs), including the reconstruction of turbulent flow fields from sparse data. However, most existing PINN formulations are deterministic and do not provide reliable quantification of epistemic uncertainty, which is critical for ill-posed problems such as data-driven Reynolds-averaged Navier-Stokes (RANS) modeling. In this work, we develop and systematically evaluate a set of probabilistic extensions of PINNs for uncertainty quantification in turbulence modeling. The proposed framework combines (i) Bayesian PINNs with Hamiltonian Monte Carlo sampling and a tempered multi-component likelihood, (ii) Monte Carlo dropout, and (iii) repulsive deep ensembles that enforce diversity in function space. Particular emphasis is placed on the role of ensemble diversity and likelihood tempering in improving uncertainty calibration for PDE-constrained inverse problems. The methods are assessed on a hierarchy of test cases, including the Van der Pol oscillator and turbulent flow past a circular cylinder at Reynolds numbers Re=3,900 (direct numerical simulation data) and Re = 10,000 (experimental particle image velocimetry data). The results demonstrate that Bayesian PINNs provide the most consistent uncertainty estimates across all inferred quantities, while function-space repulsive ensembles offer a computationally efficient approximation with competitive accuracy for primary flow variables. These findings provide quantitative insight into the trade-offs between accuracy, computational cost, and uncertainty calibration in physics-informed learning, and offer practical guidance for uncertainty quantification in data-driven turbulence modeling.

Predicting disease progression from longitudinal imaging is useful for clinical decision making and trial design. Recent methods have moved toward increasing generative complexity, but the conditions under which this complexity is necessary remain unclear. We propose that generative complexity should match the entropy of the predictable component of a task's conditional posterior, with training-inference input alignment required in all regimes. Two model-light measurements, a task-entropy analysis on raw image pairs and a posterior-concentration analysis on a stochastic model, let practitioners assess the complexity a task warrants before committing to a modeling framework. We validated this framework on a fundus autofluorescence (FAF) dataset by contrasting five conditioning configurations, sharing one architecture and training set, spanning standard conditional diffusion, inference-aligned stochastic training, and deterministic regression. Training-inference alignment produced large gains (delta-SSIM +0.082, SSIM +0.086, both p < 0.001), while the choice among aligned frameworks produced no clinically meaningful difference across evaluated metrics. Across two FAF platforms, inter-visit change was dominated by time-invariant acquisition variability rather than disease progression, and the stochastic models' posteriors collapsed to an effective point, explaining the framework equivalence. We trained a deterministic Temporal Retinal U-Net (TRU) and evaluated it on 28,899 eyes across three manufacturers and two modalities (two FAF platforms and en-face SLO), with three independent cohorts evaluated zero-shot. TRU matched or exceeded three published baselines on delta-SSIM, SSIM, and PSNR. These findings show that when disease progression is slow compared with acquisition variability, a deterministic regression model matches or outperforms more complex stochastic alternatives.

Diffusion models (DMs) have recently shown remarkable performance on inverse problems (IPs). Optimization-based methods can fast solve IPs using DMs as powerful regularizers, but they are susceptible to local minima and noise overfitting. Although DMs can provide strong priors for Bayesian approaches, enforcing measurement consistency during the denoising process leads to manifold infeasibility issues. We propose Noise-space Hamiltonian Monte Carlo (N-HMC), a posterior sampling method that treats reverse diffusion as a deterministic mapping from initial noise to clean images. N-HMC enables comprehensive exploration of the solution space, avoiding local optima. By moving inference entirely into the initial-noise space, N-HMC keeps proposals on the learned data manifold. We provide a comprehensive theoretical analysis of our approach and extend the framework to a noise-adaptive variant (NA-NHMC) that effectively handles IPs with unknown noise type and level. Extensive experiments across four linear and three nonlinear inverse problems demonstrate that NA-NHMC achieves superior reconstruction quality with robust performance across different hyperparameters and initializations, significantly outperforming recent state-of-the-art methods. The code is available at https://github.com/NA-HMC/NA-HMC.

Generative semantic communication (SemCom) harnesses pretrained generative priors to improve the perceptual quality of wireless image transmission. Existing generative SemCom receivers, however, rely on maximum a posteriori (MAP) estimation, which fundamentally cannot preserve the data distribution and thus limits achievable perceptual quality. Moreover, current diffusion-based approaches using single-domain guidance face significant limitations: latent-domain guidance is sensitive to channel noise, while image-domain guidance inherits decoder bias. Simply combining both domains simultaneously yields an overconfident pseudo-posterior. In this paper, we formulate semantic decoding as a Bayesian inverse problem and prove that posterior sampling achieves optimal perceptual quality by preserving the data distribution. Building on this insight, we propose alternating dual-domain posterior sampling (ADDPS), a diffusion-based SemCom receiver that alternately enforces latent-domain and image-domain consistency during the sampling process. This alternating strategy decomposes joint posterior sampling into simpler subproblems, avoiding gradient conflicts while retaining the complementary strengths of both domains. Experiments on FFHQ demonstrate that the proposed ADDPS achieves superior perceptual quality compared with existing methods.

Early-exit neural networks enable adaptive computation by allowing confident predictions to exit at intermediate layers, achieving 2-8$\times$ inference speedup. Despite widespread deployment, their generalization properties lack theoretical understanding -- a gap explicitly identified in recent surveys. This paper establishes a unified PAC-Bayesian framework for adaptive-depth networks. (1) Novel Entropy-Based Bounds: We prove the first generalization bounds depending on exit-depth entropy $H(D)$ and expected depth $\mathbb{E}[D]$ rather than maximum depth $K$, with sample complexity $\mathcal{O}((\mathbb{E}[D] \cdot d + H(D))/ε^2)$. (2) Explicit Constructive Constants: Our analysis yields the leading coefficient $\sqrt{2\ln 2} \approx 1.177$ with complete derivation. (3) Provable Early-Exit Advantages: We establish sufficient conditions under which adaptive-depth networks strictly outperform fixed-depth counterparts. (4) Extension to Approximate Label Independence: We relax the label-independence assumption to $ε$-approximate policies, broadening applicability to learned routing. (5) Comprehensive Validation: Experiments across 6 architectures on 7 benchmarks demonstrate tightness ratios of 1.52-3.87$\times$ (all $p < 0.001$) versus $>$100$\times$ for classical bounds. Bound-guided threshold selection matches validation-tuned performance within 0.1-0.3%.

In this paper, we proposed Bayesian Tucker decomposition (BTuD) in which residual is supposed to obey Gaussian distribution analogous to linear regression. Although we have proposed an algorithm to perform the proposed BTuD, the conventional higher-order orthogonal iteration can generate Tucker decomposition consistent with the present implementation. Using the proposed BTuD, we can perform unsupervised feature selection successfully applied to various synthetic datasets, global coupled maps with randomized coupling strength, and gene expression profiles. Thus we can conclude that our newly proposed unsupervised feature selection method is promising. In addition to this, BTuD based unsupervised FE is expected to coincide with TD based unsupervised FE that were previously proposed and successfully applied to a wide range of problems.

We propose an Expected Free Energy-based acquisition function for Bayesian optimization to solve the joint learning and optimization problem, i.e., optimize and learn the underlying function simultaneously. We show that, under specific assumptions, Expected Free Energy reduces to Upper Confidence Bound, Lower Confidence Bound, and Expected Information Gain. We prove that Expected Free Energy has unbiased convergence guarantees for concave functions. Using the results from these derivations, we introduce a curvature-aware update law for Expected Free Energy and show its proof of concept using a system identification problem on a Van der Pol oscillator. On a two-dimensional benchmark with an oscillatory landscape, our adaptive Expected Free Energy acquisition achieves competitive performance in both regret and mean squared error, unlike the typical acquisition functions that perform well in only one metric.

Nonrigid registration is conventionally divided into point set registration, which aligns sparse geometries, and image registration, which aligns continuous intensity fields on regular grids. This dichotomy is limiting for emerging scientific data such as spatial transcriptomics, where high-dimensional vector-valued functions, e.g., gene expression, are defined on irregular sparse manifolds. Researchers must therefore either sacrifice single-cell resolution through voxelization or ignore functional signals in favor of geometric alignment. We propose Domain Elastic Transform (DET), a grid-free probabilistic framework that jointly aligns geometry and function. By treating data as functions on irregular domains, DET registers high-dimensional signals directly without binning. Within a generalized Bayesian formulation, domain deformation is modeled as elastic motion guided by a joint spatial-functional likelihood. DET is fully unsupervised and scalable through registration on sampled points followed by displacement interpolation. We evaluate DET on MERFISH mouse-brain slices and Stereo-seq mouse-embryo atlases. On a 90-case MERFISH benchmark with severe perturbations and no prior initialization, DET achieved the strongest spatial overlap and topology among the evaluated pipelines, while an accelerated PASTE2 variant achieved the highest label-transfer ARI. In an atlas-scale MOSTA feasibility study without cross-stage ground truth, nonrigid refinement improved several within-pipeline anatomical-domain and boundary-consistency measures. These results suggest that grid-free function registration complements point-set, image-based, and optimal-transport approaches for high-dimensional scientific data. The DET implementation is available at https://github.com/ohirose/bcpd (since Mar, 2025).

We present a theory-guided generalized Bayesian methodology for spatio-temporal raster data, which we use to train an ensemble of stochastic feed-forward neural networks with Gaussian-distributed weights. The methodology incorporates the dependence and causal structure of a spatio-temporal Ornstein-Uhlenbeck process into training and inference by enforcing constraints on the design of the data embedding and the related optimization routine. In inference mode, the networks are employed to generate causal ensemble forecasts by applying different initial conditions at different horizons. We call this workflow MMAF-guided learning. Experiments conducted on both synthetic and real data demonstrate that our forecasts remain calibrated across multiple time horizons. Moreover, we show that on such data, shallow feed-forward architectures can achieve performance comparable to, and in some cases better than, convolutional or diffusion deep learning architectures used in probabilistic forecasting tasks.

Many disciplines need quantitative models that synthesize experimental data across multiple instances of the same general system. For example, neuroscientists must combine data from the brains of many individual animals to understand the species' brain in general. However, typical machine learning models treat one system instance at a time. Here we introduce a machine learning framework, deep probabilistic model synthesis (DPMS), that leverages system properties auxiliary to the model to combine data across system instances. DPMS specifically uses variational inference to learn a conditional prior distribution and instance-specific posterior distributions over model parameters that respectively tie together the system instances and capture their unique structure. DPMS can synthesize a wide variety of model classes, such as those for regression, classification, and dimensionality reduction, and we demonstrate its ability to improve upon single-instance models on synthetic data and whole-brain neural activity data from larval zebrafish.

Error Span Detection (ESD) is a crucial subtask in Machine Translation (MT) evaluation, aiming to identify the location and severity of translation errors. While fine-tuning models on human-annotated data improves ESD performance, acquiring such data is expensive and prone to inconsistencies among annotators. To address this, we propose a novel self-evolution framework based on Minimum Bayes Risk (MBR) decoding, named Iterative MBR Distillation for ESD, which eliminates the reliance on human annotations by leveraging an off-the-shelf LLM to generate pseudo-labels. Extensive experiments on the WMT Metrics Shared Task datasets demonstrate that models trained solely on these self-generated pseudo-labels outperform both unadapted base model and supervised baselines trained on human annotations at the system and span levels, while maintaining competitive sentence-level performance.

Machine learning forecast systems are moving beyond point predictions to full predictive distributions for future outcomes y conditional on complex inputs x. However, these distributions are often locally miscalibrated, especially for high-stakes tail events where accurate uncertainty quantification is most needed to establish trust in models. Local miscalibration occurs because training data often lack examples of low-frequency events. The goal of this paper is to describe a simple, yet flexible framework that, at deployment, produces interpretable diagnostics and a robust correction mechanism of predictive distributions when train examples are limited. With this goal in mind, we introduce a semiparametric version of the Local Amortized Diagnostic and Reshaping (LADaR) framework that posits a covariate-dependent parametric model for a diagnostic transport map regressed nonparametrically on inputs to describe how to correct tail probabilities across the feature space to match calibration data. These maps provide the user with local, real-time diagnostics and a reshaped predictive distribution that can be related back to physical processes in the input space. We apply these semiparametric diagnostic transport maps to short-term tropical cyclone intensity forecasting to detect evolutionary modes linked to local miscalibration in the National Hurricane Center's forecasts and improve predictions for severe weather hazards.

We analyze the Bayesian regret of the Gaussian process posterior sampling reinforcement learning (GP-PSRL) algorithm. Posterior sampling is a heuristic for decision-making under uncertainty that has been used to develop successful algorithms for a variety of continuous control problems. However, theoretical work on GP-PSRL is limited. All known regret bounds either have a sub-optimal growth rate, require strong smoothness assumptions, or fail to properly account for the fact that the set of possible system states is unbounded. Through a recursive application of the Borell-Tsirelson-Ibragimov-Sudakov inequality, we show that, with high probability, the states actually visited by the algorithm are contained within a ball of near-constant radius. We then use the chaining method to control the regret suffered by GP-PSRL under weak smoothness conditions. Our main result is a Bayesian regret bound of the order $\widetilde{\mathcal{O}}(H\sqrt{γ_TT})$, where $H$ is the horizon, $T$ is the number of time steps and $γ_T$ is the expected information gain. With this result, we resolve the limitations with prior theoretical work on PSRL, and provide the theoretical foundation and tools for analyzing PSRL in complex settings.

Modern tabular foundation models such as TabPFN and TabICL naturally produce full predictive distributions, while the benchmarks used to evaluate them (TabArena, TALENT, and others) still rely almost exclusively on point-estimate metrics (RMSE, $R^2$). This mismatch implicitly rewards machine learning models or pipelines that elicit a good conditional mean while ignoring the quality of the predictive distribution. We make the case for using proper scoring rules for training, fine-tuning, and benchmarking (ranking) of tabular foundation models. Although all strictly proper scoring rules are theoretically equivalent at the population level, they may differ on finite data: We demonstrate analytically and empirically that different scoring rules can induce different inductive biases during finite-sample optimization, leading to different model performance. We validate this finding by running fine-tuning experiments with TabPFN and TabICL using different scoring rules for various data sets, revealing non-trivial interactions between training objectives and evaluation metrics. Our results show that practitioners can adapt tabular foundation models to task-specific scoring objectives, and that the choice of scoring rule can influence model behavior in practice.

Theoretical and applied research into privacy encompasses an incredibly broad swathe of differing approaches, emphases and aims. This work introduces a novel quantitative notion of privacy that is both contextual and specific. Building on and extending ideas from statistical disclosure control and differential privacy, our aim is to model the implications of a disclosure decision in an adversarial setting. Our definition relies on concepts inherent to standard Bayesian decision theory, while departing from them in several important respects. In particular, (i) inference about the data itself becomes meaningful and (ii) the party controlling the release of sensitive information should make disclosure decisions from the prior viewpoint, rather than conditional on the data, which is a feature shared with Bayesian design. Illuminating toy examples are exploited towards highlighting the specificities of the method.

We consider the optimization problem of an expensive-to-evaluate black-box function, in which we can obtain noisy function values in parallel. For this problem, parallel Bayesian optimization (PBO) is a promising approach, which aims to optimize with fewer function evaluations by selecting a diverse input set for parallel evaluation. However, existing PBO methods suffer from poor practical performance or lack theoretical guarantees. In this study, we propose a PBO method, called randomized kriging believer (KB), based on a well-known KB heuristic and inheriting the advantages of the original KB: low computational complexity, a simple implementation, versatility across various BO methods, and applicability to asynchronous parallelization. Furthermore, we show that our randomized KB achieves Bayesian expected regret guarantees. We demonstrate the effectiveness of the proposed method through experiments, including those on real-data emulators.

Generalization bounds for deep learning models are typically vacuous, not computable or restricted to specific model classes. In this paper, we tackle these issues by providing new disagreement-based certificates for the gap between the true risk of any two predictors. We then bound the true risk of the predictor of interest via a surrogate model that enjoys tight generalization guarantees, and by evaluating our disagreement bound on an unlabeled dataset.We empirically demonstrate the tightness of the obtained certificates and showcase the versatility of the approach by training surrogate models leveraging three different frameworks: sample compression, model compression and PAC-Bayes theory. Importantly, such guarantees are achieved without modifying the target model, nor adapting the training procedure to the generalization framework.

Test-time guidance is a widely used mechanism for steering pretrained diffusion models toward outcomes specified by a reward function. Existing approaches, however, focus on maximizing reward rather than sampling from the true Bayesian posterior, leading to miscalibrated inference. In this work, we show that common test-time guidance methods do not recover the correct posterior distribution and identify the structural approximations responsible for this failure. We then propose consistent alternative estimators that enable calibrated sampling from the Bayesian posterior. We significantly outperform previous methods on a set of Bayesian inference tasks, and set a new state-of-the-art PSNR in black hole image reconstruction.

In safety-critical classification, the cost of failure is often asymmetric, yet Bayesian deep learning summarises epistemic uncertainty with a single scalar, mutual information (MI), that cannot distinguish whether a model's ignorance involves a benign or safety-critical class. We decompose MI into a per-class vector $C_k(x)=σ_k^{2}/(2μ_k)$, with $μ_k{=}\mathbb{E}[p_k]$ and $σ_k^2{=}\mathrm{Var}[p_k]$ across posterior samples. The decomposition follows from a second-order Taylor expansion of the entropy; the $1/μ_k$ weighting corrects boundary suppression and makes $C_k$ comparable across rare and common classes. By construction $\sum_k C_k \approx \mathrm{MI}$, and a companion skewness diagnostic flags inputs where the approximation degrades. After characterising the axiomatic properties of $C_k$, we validate it on three tasks: (i) selective prediction for diabetic retinopathy, where critical-class $C_k$ reduces selective risk by 34.7% over MI and 56.2% over variance baselines; (ii) out-of-distribution detection on clinical and image benchmarks, where $\sum_k C_k$ achieves the highest AUROC and the per-class view exposes asymmetric shifts invisible to MI; and (iii) a controlled label-noise study in which $\sum_k C_k$ shows less sensitivity to injected aleatoric noise than MI under end-to-end Bayesian training, while both metrics degrade under transfer learning. Across all tasks, the quality of the posterior approximation shapes uncertainty at least as strongly as the choice of metric, suggesting that how uncertainty is propagated through the network matters as much as how it is measured.

Electroencephalography (EEG)-based P300 brain-computer interfaces (BCIs) enable communication without physical movement by detecting stimulus-evoked neural responses. Accurate and efficient decoding remains challenging due to high dimensionality, temporal dependence, and complex interactions across EEG channels. Existing approaches often treat channels independently or rely on black-box models, limiting interpretability and personalization. We propose a sparse Bayesian time-varying regression framework that explicitly models pairwise EEG channel interactions while performing temporal feature selection, where a relaxed-thresholded Gaussian process prior induces structured sparsity in both channel-specific and interaction effects, enabling interpretable identification of task-relevant channels and channel pairs. Applied to a public P300 speller dataset of 55 participants, our method achieves a 96.4% median character-level accuracy using all sequence repetitions and attains the best overall decoding performance among competing statistical and deep learning approaches. Incorporating channel interactions yields subgroup-specific gains in character-level accuracy, particularly among participants abstained from alcohol (median 7%, up to 14%). Importantly, our approach improves median BCI-Utility by more than 10% at its optimal operating point, achieving peak throughput after only six sequence repetitions. These results demonstrate that explicitly modeling structured EEG channel interactions within a principled Bayesian framework enhances accuracy and user-centric throughput, and supports personalization in P300 BCI systems.

Bayesian quadrature is a probabilistic, model-based approach to numerical integration, the estimation of intractable integrals, or expectations. Although Bayesian quadrature was popularised already in the 1980s, no systematic and comprehensive treatment has been published. The purpose of this survey is to fill this gap. We review the mathematical foundations of Bayesian quadrature from different points of view; present a systematic taxonomy for classifying different Bayesian quadrature methods along the three axes of modelling, inference, and sampling; collect general theoretical guarantees; and provide a controlled numerical study that explores and illustrates the effect of different choices along the axes of the taxonomy. We also provide a realistic assessment of practical challenges and limitations to application of Bayesian quadrature methods and include an up-to-date and nearly exhaustive bibliography that covers not only machine learning and statistics literature but all areas of mathematics and engineering in which Bayesian quadrature or equivalent methods have seen use.

We theoretically justify the recent empirical finding of [Teh et al., 2025] that a transformer pretrained on synthetically generated data achieves strong performance on empirical Bayes (EB) problems. We take an indirect approach to this question: rather than analyzing the model architecture or training dynamics, we ask why a pretrained Bayes estimator, trained under a prespecified training distribution, can adapt to arbitrary test distributions. Focusing on Poisson EB problems, we identify the existence of universal priors such that training under these priors yields a near-optimal regret bound of $\widetilde{O}(\frac{1}{n})$ uniformly over all test distributions. Our analysis leverages the classical phenomenon of posterior contraction in Bayesian statistics, showing that the pretrained transformer adapts to unknown test distributions precisely through posterior contraction. This perspective also explains the phenomenon of length generalization, in which the test sequence length exceeds the training length, as the model performs Bayesian inference using a generalized posterior.

This article addresses the issue of estimating observation parameters (response and error parameters) in inverse problems. The focus is on cases where regularization is introduced in a Bayesian framework and the prior is modeled by a diffusion process. In this context, the issue of posterior sampling is known to be thorny, and a recent paper proposes a notably simple and effective solution. Additionally, it opens an remarkable flexibility when it comes to estimating observation parameters. The proposed strategy enables to define an optimal estimator for both observation parameters and image of interest. Furthermore, the strategy provides a means for uncertainty quantification. In addition, MCMC algorithms allow for the computation of estimates and properties of posteriors, while offering some guarantees. The paper presents several numerical experiments that clearly confirm the computational efficiency and the quality of both estimates and uncertainty quantification.

This paper addresses the issue of inversion in cases where (1) the observation system is modeled by a linear transformation and additive error, (2) the problem is ill-posed and regularization relies on a Bayesian strategy, (3)~the prior is modeled by a diffusion process adjusted on an available large set of examples. In this context, it is known that the issue of posterior sampling is a thorny one and the paper introduces a Gibbs algorithm. It appears that this avenue has not been explored, and we show that it is particularly effective and remarkably simple. In addition, it provides clear elements regarding convergence guarantees in a specific case and arguments supporting such guarantees in practical cases. The results are clearly confirmed by numerical simulations based on a toy example.

In many experimental settings, active learning can improve sample efficiency by sequentially selecting where to measure, which is particularly valuable when experiments are expensive. Gaussian processes with variance-based acquisition criteria are widely used for this purpose, but have two limitations. First, they are observation-independent: their posterior variance depends only on where samples are acquired, not on what is measured, impairing their sensitivity to the structure of the acquired data. Second, they inflate the variance near boundaries, leading to excessive sampling at the edges of the space compared to the interior. These limitations undermine the gains in sampling efficiency expected from sequential acquisition. We address both limitations. We derive a reconstruction-driven design density and use the posterior mean to build a training-free warp that places more measurements where the target function varies rapidly. A geometric equalizer separately corrects boundary bias. Across sixteen synthetic and two real-data benchmarks, the geometric equalizer consistently improves function reconstruction by correcting boundary bias, while the reconstruction warp provides further gains by concentrating measurements where the posterior mean varies rapidly.

We introduce a new multivariate statistical problem that we refer to as the Ensemble-conditioned Inverse Problem (EIP). The aim of EIP is to invert for an ensemble that is distributed according to the pushforward of a prior under a forward process. In high energy physics (HEP), this is related to a widely known problem called unfolding, which aims to reconstruct the true physics distribution from observations that are distorted by detector effects. The EIP also arises in full waveform inversion (FWI) and inverse imaging with unknown priors. We propose non-iterative inference-time methods that construct posterior samplers based on a new class of conditional generative models, which we call ensemble inverse generative models. For the posterior modeling, these models additionally use the ensemble information contained in the observation set on top of single observations. Unlike existing methods, our proposed methods avoid explicit and iterative use of the forward model at inference time via training across several sets of truth-observation pairs that are consistent with the same forward model, but originate from a wide range of priors. We empirically demonstrate that this training procedure can implicitly encode the likelihood model, enabling direct posterior inference for unseen priors to some degree. We benchmark the proposed method on several synthetic and real datasets in inverse imaging, HEP, and FWI. Our code is available at https://github.com/ZhengyanHuan/EIP.

Evolutionary dynamic optimization (EDO) commonly assumes that environmental changes can be detected from fitness variations and handled through random re-initialization, historical solutions, or learned transition patterns. Online calibration of black-box simulators introduces a different setting, where the dynamic objective is induced by sequential observations and a changing calibration window, rather than being controlled by explicit variables. Fitness variations therefore cannot be directly attributed to regime changes, while the unknown relationship between successive regimes limits conventional adaptation. We formulate this setting as an observation-driven dynamic optimization problem and propose PosEDO, which augments fitness-based EDO with an observation-conditioned parameter-space signal. PosEDO learns this signal online as a posterior distribution over simulator parameters from parameter-trajectory pairs generated during evolutionary evaluation, using posterior shifts for change detection and posterior samples for population adaptation. The new evaluation records are further utilized for online posterior updating without additional simulator calls. Experiments on nonstationary economic and financial simulators show that PosEDO improves calibration accuracy, optimization performance, and change-detection quality over representative EDO baselines.

Since the turn of the century, approximate Bayesian inference has steadily evolved as new computational techniques have been incorporated to handle increasingly complex, large-scale predictive problems. The recent success of deep neural networks and foundation models has now given rise to a new paradigm in statistical modeling, in which Bayesian inference can be amortized through large-scale learned predictors. In amortized inference, substantial computation is required at the beginning to train a neural network, but it can subsequently produce approximate posteriors or predictions at much lower computational cost across a wide range of tasks. While the typical Bayesian inference procedures are computationally expensive due to repeated likelihood calculations and Monte Carlo steps for each new dataset, amortized inference provides a much lower computational cost at deployment. Despite the growing popularity of amortized inference, its statistical interpretation and position within Bayesian inference remain poorly explored. In this paper, we present a statistical perspective on several major neural architectures, including feedforward networks, Deep Sets, and Transformers, and examine how they naturally support amortized Bayesian inference. We explore how these models perform structured approximation and also probabilistic reasoning in ways that yield controlled generalization error throughout a wide range of deployment scenarios, and how these properties can be harnessed for Bayesian computation. Via simulation studies, we evaluate the accuracy, robustness, and uncertainty quantification of amortized inference across varying sample sizes, varying noise distributional families, varying sparsity levels, and multimodality, highlighting its strengths and limitations.

We prove that Thompson sampling exhibits $\tilde{O}(σd \sqrt{T} + d r \sqrt{\mathrm{Tr}(Σ_0)})$ Bayesian regret in the linear-Gaussian bandit with a $\mathcal{N}(μ_0, Σ_0)$ prior distribution on the coefficients, where $d$ is the dimension, $T$ is the time horizon, $r$ is the maximum $\ell_2$ norm of the actions, and $σ^2$ is the noise variance. In contrast to existing regret bounds, this shows that to within logarithmic factors, the prior-dependent burn-in'' term $d r \sqrt{\mathrm{Tr}(Σ_0)}$ decouples additively from the minimax (long run) regret $σd \sqrt{T}$. Previous regret bounds exhibit a multiplicative dependence on these terms. We establish these results via a new elliptical potential'' lemma, and also provide a lower bound indicating that the burn-in term is unavoidable.

Knowledge Distillation (KD) is a central paradigm for transferring knowledge from a large teacher network to a typically smaller student model, often by leveraging soft probabilistic outputs. While KD has shown strong empirical success in numerous applications, its theoretical underpinnings remain only partially understood. In this work, we adopt a Bayesian perspective on KD to rigorously analyze the convergence behavior of students trained with Stochastic Gradient Descent (SGD). We study two regimes: $(i)$ when the teacher provides the exact Bayes Class Probabilities (BCPs); and $(ii)$ supervision with noisy approximations of the BCPs. Our analysis shows that learning from BCPs yields variance reduction and removes neighborhood terms in the convergence bounds compared to one-hot supervision. We further characterize how the level of noise affects generalization and accuracy. Motivated by these insights, we advocate the use of Bayesian deep learning models, which typically provide improved estimates of the BCPs, as teachers in KD. Consistent with our analysis, we experimentally demonstrate that students distilled from Bayesian teachers not only achieve higher accuracies (up to +4.27%), but also exhibit more stable convergence (up to 30% less noise), compared to students distilled from deterministic teachers.

Neural networks trained with standard objectives exhibit behaviors characteristic of probabilistic inference: soft clustering, prototype specialization, and Bayesian uncertainty tracking. These phenomena appear across architectures -- in attention mechanisms, classification heads, and energy-based models -- yet existing explanations often rely on loose analogies to mixture models or post-hoc architectural interpretation. We provide a direct explanation. For any objective with log-sum-exp structure over distances or energies, the gradient with respect to each distance is exactly the negative posterior responsibility of the corresponding component: $\partial L / \partial d_j = -r_j$. The identity is algebraic, requiring only differentiability; it is a specialization of Fisher's identity, and its significance here is its address: standard neural objectives instantiate it without modification. The consequence is that gradient descent on such objectives performs generalized expectation-maximization implicitly, with responsibilities arising as gradients to be applied rather than auxiliary variables to be computed. This result unifies three regimes of learning: unsupervised mixture modeling, where responsibilities are fully latent; attention, where responsibilities are conditioned on queries; and cross-entropy classification, where supervision clamps responsibilities to targets. Our claims live at training time: the responsibility-weighted gradient dynamics recently documented in transformers follow from the objective's geometry. The in-context Bayesian computation that trained transformers perform at inference time is the endpoint of these dynamics, not their per-step content.

We consider problems of parameter estimation where design variables can be actively optimized to maximize information gain. To this end, we introduce JADAI, a framework that jointly amortizes Bayesian adaptive design and inference by training a policy, a history network, and an inference network end-to-end. The networks minimize a generic loss that aggregates incremental reductions in posterior error along experimental sequences. Inference networks are instantiated with diffusion-based posterior estimators that can approximate high-dimensional and multimodal posteriors at every experimental step. Across standard adaptive design benchmarks, JADAI achieves superior or competitive performance.

Diffusion models have recently emerged as powerful learners for simulation-based inference (SBI), enabling fast and accurate estimation of latent parameters from simulated and real data. Their score-based formulation offers a flexible way to learn conditional or joint distributions over parameters and observations, thereby providing a versatile solution to various modeling problems. In this tutorial review, we synthesize recent developments on diffusion models for SBI, covering design choices for training, inference, and evaluation. We highlight opportunities created by various concepts such as guidance, score composition, flow matching, consistency models, and joint modeling. Furthermore, we discuss how efficiency and statistical accuracy are affected by noise schedules, parameterizations, and samplers. Finally, we illustrate these concepts with case studies across parameter dimensionalities, simulation budgets, and model types, and outline open questions for future research.

This work shows how adaptivity can enhance value realization of digital twins in civil engineering. We focus on adapting the state transition models within digital twins represented through probabilistic graphical models. The bi-directional interaction between the physical and virtual domains is modeled using dynamic Bayesian networks. By treating state transition probabilities as random variables endowed with conjugate priors, we enable hierarchical online learning of transition dynamics from a state to another through effortless Bayesian updates. We provide the mathematical framework to account for a larger class of distributions with respect to the current literature on digital twins. To compute dynamic policies with precision updates we solve parametric Markov decision processes through reinforcement learning. The proposed adaptive digital twin framework enjoys enhanced personalization, increased robustness, and improved cost-effectiveness. We assess our approach on a case study involving structural health monitoring and maintenance planning of a railway bridge.

Diffusion models provide powerful generative priors for solving inverse problems by sampling from a posterior distribution conditioned on corrupted measurements. Existing methods primarily follow two paradigms: direct methods, which approximate the likelihood term, and proximal methods, which incorporate intermediate solutions satisfying measurement constraints into the sampling process. Under standard Gaussian approximations and locally-linear measurements, we demonstrate that these approaches differ fundamentally in their treatment of the diffusion denoiser's Jacobian within the likelihood term. While this Jacobian encodes critical prior knowledge of the data distribution, training-induced non-idealities can degrade performance in zero-shot settings. In this work, we bridge direct and proximal approaches by proposing a principled Jacobian-Aware Posterior Sampler (JAPS). JAPS leverages the Jacobian's prior knowledge while mitigating its detrimental effects through a corresponding proximal solution, requiring no additional computational cost. Additionally, we integrate our guidance into DDIM sampling, with a corrected conditional factor that has been missing in previous works. Our method enhances reconstruction quality across diverse linear and nonlinear noisy imaging tasks, outperforming existing diffusion-based baselines in perceptual quality while maintaining or improving distortion metrics.

From a Bayesian perspective, score-based diffusion solves inverse problems through joint inference, embedding the likelihood with the prior to guide the sampling process. However, this formulation fails to explain its practical behavior: the prior offers limited guidance, while reconstruction is largely driven by the measurement-consistency term, leading to an inference process that is effectively decoupled from the diffusion dynamics. We show that the diffusion prior in these solvers functions primarily as a warm initializer that places estimates near the data manifold, while reconstruction is driven almost entirely by measurement consistency. Based on this observation, we introduce \textbf{DAPS++}, which fully decouples diffusion-based initialization from likelihood-driven refinement, allowing the likelihood term to guide inference more directly while maintaining numerical stability and providing insight into why unified diffusion trajectories remain effective in practice. By requiring fewer function evaluations (NFEs) and measurement-optimization steps, \textbf{DAPS++} achieves high computational efficiency and robust reconstruction performance across diverse image restoration tasks.

Clinical language models often assign high confidence to incorrect predictions, particularly in high-severity and out-of-distribution cases. We present MedBayes-Lite, a retraining-free uncertainty governance layer for transformer-based clinical predictors. It combines Monte Carlo dropout, predictive calibration, and confidence-guided abstention to defer low-confidence predictions for human review, adding no trainable parameters. Evaluated on MedMCQA and MedQA-USMLE, MedBayes-Lite reduces expected calibration error by 0.23 to 0.33 and drives harmful overconfident errors (confident, incorrect, high-severity predictions) toward zero. Under domain shift from MedMCQA to MedQA-USMLE, it reduces confident high-severity errors from about 21% to near zero while roughly halving calibration drift. We also introduce the Clinical Uncertainty Score (CUS), which strongly correlates with harmful overconfidence (r approximately 0.88). Although the framework does not improve risk-coverage ranking, and temperature scaling or deep ensembles may provide advantages in calibration cost or risk ranking, MedBayes-Lite offers a practical calibration-and-abstention layer that reduces confident high-severity errors in clinical question-answering benchmarks.

Efficient Gaussian process (GP) inference is critical for sequential decision-making tasks such as active learning, online prediction, and Bayesian optimization. Iterative approaches of approximating the GP posterior using solvers like conjugate gradients, stochastic gradient descent, or alternating projections avoid cubic costs, but often require many iterations to converge, limiting their efficacy when the posterior is updated frequently with new data. To address this, we introduce three warm-start strategies that exploit solutions of smaller linear systems to substantially speed-up convergence when updating the posterior with new data. Our methods are supported by theoretical analysis showing reduced initialization error in reproducing kernel Hilbert space (RKHS) distance, and by empirical results on regression benchmarks and Bayesian optimization tasks. Across solvers, warm-starting achieves speed-ups of up to 19x when solving to tolerance, and produces more accurate posterior estimates under fixed compute budgets, directly improving optimization performance. These results establish warm-starting as a simple, effective, and broadly applicable tool for scaling Gaussian processes in sequential settings.

Like visual processing, language processing is susceptible to illusions in which people systematically misperceive stimuli. In one such case--the comparative illusion (CI), e.g., More students have been to Russia than I have--comprehenders tend to judge the sentence as acceptable despite its underlying nonsensical comparison. Prior research has argued that this phenomenon can be explained as Bayesian inference over a noisy channel: the posterior probability of an interpretation of a sentence is proportional to both the prior probability of that interpretation and the likelihood of corruption into the observed (CI) sentence. Initial behavioral work has supported this claim by evaluating a narrow set of alternative interpretations of CI sentences and showing that comprehenders favor interpretations that are more likely to have been corrupted into the illusory sentence. In this study, we replicate and go substantially beyond this earlier work by directly predicting the strength of illusion with a quantitative model of the posterior probability of plausible interpretations, which we derive through a novel synthesis of statistical language models with human behavioral data. Our model explains not only the fine gradations in the strength of CI effects, but also a previously unexplained effect caused by pronominal vs. full noun phrase than-clause subjects. These findings support a noisy-channel theory of sentence comprehension by demonstrating that the theory makes novel predictions about the comparative illusion that bear out empirically. This outcome joins related evidence of noisy channel processing in both illusory and non-illusory contexts to support noisy channel inference as a unified computational-level theory of diverse language processing phenomena.