Bayesian

We introduce a theoretical framework for analyzing federated learning in a generative setting through a teacher-multiple interacting students scenario, in which each student receives a distinct realization of the data, either through a different noise corruption or by accessing a different subset, possibly of varying size. Using theoretical tools in equilibrium disordered system, we analytically show that interactions among students systematically enhance learning performance: highly noisy students require fewer samples to recover the underlying pattern, while low-noise students achieve a larger overlap with the ground-truth signal. We derive the optimal Bayesian conditions for teacher recovery as functions of the sample complexity, noise level, and interaction strength, and validate these predictions through numerical simulations. The resulting dynamics can be mapped onto equilibrium sampling in a Restricted Boltzmann Machine with a structured hidden layer, providing a principled theoretical understanding of how interactions improve distributed generative modeling.

ML classifiers deployed in high-stakes domains produce predictions whose quality varies systematically across subgroups. For granular subgroups defined by intersections of multiple features, predictions are often inconsistent with the observed data: the model's outputs contradict the evidence available for that subgroup. This problem is exacerbated by regularisation, which improves aggregate performance by collapsing small subgroups into larger groups, disproportionately affecting demographic minorities. We define two requirements for consistent prediction: determinism (identical individuals receive identical predictions) and statistical consistency (we cannot reject, at significance level alpha, the hypothesis that the predictions for a subgroup were drawn from the Bayesian optimal target distribution inferred for that subgroup). From these requirements we derive the Fair Bayesian classifier, which enforces both across every group and subgroup simultaneously and abstains whenever no consistent deterministic prediction is possible. On three benchmark datasets (Adult, COMPAS, and Bank Marketing), standard classifiers produce statistically inconsistent predictions for a substantial proportion of subgroups. Our classifier achieves zero consistency error by construction while exceeding baseline accuracy and multicalibration on every dataset tested. Statistical consistency provides a principled foundation for prediction quality with direct implications for algorithmic fairness. Minority demographics are disproportionately concentrated in small subgroups, precisely where frequentist inference is least reliable; addressing this inference problem is therefore a necessary step toward fair ML. By enforcing Bayesian consistency at the finest resolution the data supports, the our classifier demonstrates that exhaustive subgroup fairness with principled abstention is achievable in practice.

Attention is the key mechanism underlying in-context learning in transformers, and attention patterns have been observed empirically to emerge abruptly during training. We present a Bayesian theory of feature learning in attention; we then focus on how the copy subcircuit in the first layer of an induction head is learned by analyzing a single-layer softmax attention network trained on a copy task. We derive a closed-form posterior over the attention matrix and reduce it to a low-dimensional order parameter space. This reduction reveals a phase transition in the amount of training data, which we verify using both Bayesian sampling and standard training with Adam. We contrast our results with linear attention and find that softmax attention exhibits a \emph{first-order phase transition} while in linear attention an initial \emph{second-order phase transition} is followed by a smooth, continuous evolution toward the structured attention pattern (\emph{crossover}). Our work provides a first-principles theoretical account of the abrupt emergence of the copy subcircuit, reminiscent of the one observed in training large language models.

Physics-informed neural particle flow (PINPF) learns a deterministic transport field that moves particles from a prior distribution toward a Bayesian posterior while enforcing the governing probability-evolution equation. However, the standard PINPF velocity model processes particles independently and therefore does not explicitly condition its transport decisions on the empirical particle population. This paper introduces population-aware PINPF (PA-PINPF), which augments each particle update with a permutation-invariant Deep Sets representation of the full particle set. We investigate two population encoders. PA-PINPF-State summarizes the particle states, whereas PA-PINPF-Feature summarizes the complete local physics-informed feature vectors, including particle position, pseudo-time, measurement information, likelihood values, and score information. The latter allows the population context to represent not only particle-cloud geometry, but also the population-level Bayesian transport geometry. The methods retain the original unsupervised physics-informed residual objective and require no ground-truth posterior samples during training. Experiments on range-measurement tasks and nonlinear time-difference-of-arrival posterior transport demonstrate that both population-aware variants improve over particle-wise PINPF, while feature-population encoding provides the strongest performance. These results show that population-level physics features provide useful global information for learned Bayesian particle transport.

Large-scale SLAM remains challenging due to accumulated trajectory drift and the increasing computational cost of maintaining global consistency. Submap joining alleviates these issues by constructing locally consistent submaps and subsequently fusing them into a global map. However, existing occupancy-based submap joining methods operate on discrete grids, resulting in non-smooth gradients during optimization and neglecting the uncertainty associated with occupancy estimates. We propose the first continuous probabilistic submap joining framework that jointly optimizes submap poses and a global occupancy field in the latent log-odds space. The framework employs an information-preserving sparse Bayesian formulation that compresses raw occupancy observations into sufficient-statistic log-odds tuples while retaining the posterior information of the original observations. This yields closed-form predictive mean and variance estimates for occupancy mapping, which directly enable a submap joining formulation with analytical Jacobians, leading to more accurate submap joining and yielding a closed-form optimal global map upon pose convergence. Experiments on both simulated and large-scale real-world datasets demonstrate that the proposed method achieves higher pose accuracy and improved global consistency than state-of-the-art grid-based submap joining approaches, while producing more compact map representations and better-calibrated uncertainty estimates than existing continuous occupancy mapping methods.

Bayesian optimization (BO) is a central tool for sample-efficient design, and latent-space Bayesian optimization (LSBO) extends it to structured objects such as molecules and proteins. In parallel, tabular foundation models such as TabPFN and TabICL now achieve state-of-the-art regression performance and are increasingly used as BO surrogates. Because their Bayesian behavior is induced by large synthetic pretraining collections, the composition of this pretraining distribution is crucial. LSBO creates a distinctive mismatch: the induced map from latent code to objective value differs markedly from the regression tasks used to train current in-context models. We address this mismatch by complementing the pretraining stage of tabular foundation model surrogates with synthetic optimization tasks defined on the latent space of a molecular VAE. The continued-pretraining objective features a regularizer that anchors the model to the original checkpoint, preserving its broad regression prior while avoiding overspecialization to the adaptation tasks. On held-out molecular optimization benchmarks, the resulting model achieves strong performance, supporting the relevance of LSBO-specific adaptation for in-context surrogates.

Wastewater influenza surveillance can reveal community circulation before clinical reporting, but wastewater alone is not a fully identifiable proxy for human burden. Existing wastewater models assume a fixed evidence set, while generic evidence-acquisition methods treat official surveillance streams as interchangeable costly features. We cast wastewater-first influenza monitoring as a selective decision problem: starting from mandatory wastewater evidence, the system must decide whether wastewater is sufficient, which delayed official stream to query next, and when abstention is the only scientifically defensible action under source ambiguity. We propose Bayesian Selective Latent Inference (BSLI), a principled Bayesian method that maintains a posterior over latent burden and identifiability, certifies answerability through explicit scientific gates, and optimizes query-stop decisions with an exact cost-calibrated Bellman policy. We prove the key variational, answerability, Bellman-optimality, and one-dimensional cost-calibration properties. On a fixed public-data benchmark with 5,933 forecasting episodes and 3,102 source-ambiguity episodes, BSLI improves the matched-budget cost-performance frontier while preserving conservative abstention under source ambiguity.

Accurate calibration of particle-based simulators is crucial for robotic earthwork simulation, but analytical calibration is challenging due to this task's highly nonlinear particle dynamics and the black-box nature of conventional simulators. Although simulation-based inference (SBI) can estimate posterior distributions over simulation parameters solely from forward simulations, applying SBI directly to high-fidelity (HF) particle simulators is often computationally prohibitive. Low-fidelity (LF) simulators with coarser particles can reduce this cost, but changes in particle size and particle count shift the parameter values needed to reproduce the same observation, producing biased LF posteriors. We propose Bridged SBI, which leverages a biased but informative LF posterior to guide HF inference. This method first uses inexpensive LF simulations to identify a coarse high-density parameter region, and then it learns a local residual bridge to transport LF posterior samples toward HF-consistent regions by correcting the LF--HF discrepancy. We analyze how sequential multi-fidelity SBI (Naive-MF) can suffer from LF-induced posterior miscoverage when it directly relies on the LF posterior without discrepancy correction. We then show that Bridged SBI is designed to alleviate this issue by explicitly modeling the LF--HF discrepancy through residual correction. Experiments on both sim-to-sim particle-parameter calibration and real-to-sim calibration with real soil observation show that Bridged SBI produces more accurate and reliable HF posteriors than HF-only SBI or the Naive-MF baseline, especially under limited HF simulation costs.

Modern data-driven applications increasingly involve learning from multiple heterogeneous sources, where a target dataset is limited but related information is available across domains. Naively combining these sources can degrade performance when relevance varies or spurious signals are present, posing a fundamental challenge for trustworthy cross-domain learning. We propose Projection Transfer Learning (ProjectionTL), a unified framework that integrates hierarchical Bayesian modeling with adaptive projection for selective knowledge transfer. The key idea is to decouple transfer at two levels: first, we construct a source-guided hierarchical prior that aggregates information across sources using data-driven weights, capturing global alignment between each source and the target; second, we refine this borrowing through a posterior-projection step that operates at the feature level, selectively retaining coordinates that exhibit local agreement with the target signal. This two-stage design enables the method to simultaneously perform source selection and feature selection, thereby mitigating negative transfer while preserving interpretability. ProjectionTL provides a principled approach to integrating heterogeneous data across domains, bridging statistical modeling and modern machine learning paradigms for robust and interpretable transfer. Through simulations and real-world biomedical applications, we demonstrate improved accuracy, stability, and interpretability compared to existing methods. Our framework offers a scalable and generalizable strategy for trustworthy cross-domain learning in high-dimensional settings.

Bayesian optimization (BO) is a widely used approach for black-box optimization that uses a Gaussian process (GP) as a surrogate and guides sequential evaluations via an acquisition function, with the ultimate goal of locating the global optimum $\mathbf{x}^{\star}$. To align with this goal, information-based acquisition functions such as Predictive Entropy Search (PES) model $\mathbf{x}^{\star}$ as a random variable and reduce the entropy of its distribution, but approximating this distribution via traditional GP posterior sampling is computationally expensive. To address this limitation, we leverage Conditional Diffusion Models (CDMs) to efficiently approximate the distribution of $\mathbf{x}^{\star}$ and develop BO-inherent training strategies for CDMs. Motivated by the structural properties of the CDM-learned distribution, we further develop an acquisition strategy termed Diffusion-based Mode Seeking (DMS) to guide the sequential evaluation. We establish a sub-optimality guarantee for the CDM-learned distribution and demonstrate through extensive experiments that DMS outperforms standard BO baselines.

Classical sparse Type-II Bayesian methods for M/EEG brain imaging support joint estimation of source and noise hyperparameters, but rely on fixed iterative update rules. Although these updates are principled and interpretable, their dynamics cannot be adapted from data. We propose to learn the update mechanism itself while preserving the underlying Bayesian structure by unfolding a classical joint hyperparameter-learning solver into a trainable neural architecture whose layers mirror the original iterations. The resulting framework is initialized to recover the classical solver exactly before training and is enriched through progressively more expressive correction-learning mechanisms, ranging from learnable biases to adaptive MLP and attention-based contextual refinements. In this way, training does not replace Bayesian inference with a black-box predictor, but instead learns structured correction terms while retaining the interpretability and model-based character of the original update dynamics. Structured correction learning therefore aims to improve empirical reconstruction performance without replacing the original model-based inference mechanism. Experimental results show that the learned correction variants improve reconstruction performance and convergence behavior over the baseline unfolded solver while preserving its algorithmic transparency.

Finetuning data selection requires balancing two competing goals: selecting examples that improve the downstream objective, and doing so without repeatedly finetuning models. Train-free selectors are scalable but rely on proxies such as embedding similarity or clustering, which may not match the target objective. Train-based selectors better reflect downstream utility through gradient signals, subset evaluation, or Shapley attribution, but require many costly train--evaluate iterations. We propose Hierarchical Active Region Pruning (HARP), an efficient train-based selector that preserves downstream alignment while reducing selection cost. HARP organizes the training pool into a node--leaf hierarchy, evaluates only representative leaves, and infers unmeasured utilities with empirical Bayes posteriors. It then selects data using two complementary envelopes: HARP-C, which conservatively controls redundancy, and HARP-E, which additively rewards complementary regions. We theoretically show that, under local smoothness and bounded estimation error, HARP controls selection error while reducing train--evaluate cost. We further validate that HARP variants achieve the best result and outperform the strongest baseline by up to $+8.9$ points, while using roughly $7\times$ fewer training examples.

Representative data is fundamental in machine learning, as limited data hinders generalisation. Collecting sufficient real-world samples is often infeasible. Synthetic data generation offers a practical solution, but only if the generated data faithfully reflects the structure of real observations. In this paper, a method for generating synthetic regression datasets that structurally resemble physics equations from a given equation corpus is presented. The approach uses a Bayesian Probabilistic Context-Free Grammar to capture the underlying algebraic structure of the corpus, from which novel equations are sampled. To ensure the generated inputs lie within a physically meaningful domain, the applicability domain is characterised for each equation through non-intrusive probing, also recovering inter-variable constraints. Input sampling further mimics realistic experimental conditions by drawing from random sub-ranges of the valid domain with mixed uniform and truncated normal distributions. The generated data is statistically validated against the Feynman equation corpus using Kolmogorov-Smirnov tests. The generated equations match the corpus on all of the eight studied structural features, compared to only two for an unsmoothed purely probabilistic grammar, demonstrating that the Bayesian prior is essential for structural fidelity given the size of the corpus. In a downstream hyperparameter-tuning task, a gradient-boosted regressor tuned on the synthetic data picks, on average, the 6th-best configuration out of 20 on real data, matching the result of tuning on real data itself and substantially outperforming random expression trees (10th) and noise (19th).

Whilst the vulnerability of graph neural networks (GNNs) to adversarial attacks poses a critical threat to graph representation learning, the understanding of the robust generalization behavior remains a fundamental challenge in the adversarial setting. Recently, PAC-Bayesian margin-based generalization analysis substantially advances this line of research by providing a flexible and data-dependent analytical framework. However, existing robust analyses often rely on isotropic Gaussian posteriors and control weight perturbations in the full parameter space, which limits the ability to capture heterogeneous parameter sensitivity yet hinges on hidden-width-dependent complexity terms, resulting in not-tight-enough generalization bounds. In this paper, we extend a recently proposed sensitivity-aware PAC-Bayesian framework from deep neural networks to message passing GNNs (MPGNNs) and derive a tighter robust generalization bound in the adversarial setting. Specifically, we first quantify how sensitive the perturbations across different parameter blocks are to the network outputs by deriving the output Jacobians with respect to the weight parameters. Exploiting the fact that these Jacobian matrices have rank at most $K$ in $K$-class graph classification, we then construct Jacobian-aligned sensitivity matrices and use anisotropic Gaussian posteriors with optimized covariances to upper bound the KL divergence in a tight way. Notably, by refining the spectral-norm dependence on the learned weights and reducing the leading dimension factor from hidden-width-dependent terms to the number of classes $K$, our analysis yields much tighter robust generalization guarantees for MPGNNs, thereby guiding their designs to enhance adversarial robustness.

Causal representation learning aims to infer the high-level latent causal concepts that give rise to observed low-level measurements. This is particularly relevant for heterogeneous data from different environments or domains since distribution shifts often arise through sparse, localized changes in some of the underlying causal mechanisms, while other parts of the generative process remain unchanged. Whereas identifiability of causal representations has been studied extensively, practical uncertainty-aware methods and real-world use cases remain less explored. In this work, we propose a Bayesian approach to learning causal representations from multi-environment data, focusing on the case of discrete causal concepts and unknown multi-node soft interventions. To this end, we translate causal assumptions and interpretability desiderata into suitable priors and parametric choices within a hierarchical model. We then devise an inference scheme based on sequential Monte Carlo sampling to approximate the resulting multimodal posterior. We showcase our approach through case studies on social survey data, where latent causal concepts correspond to cultural values or political opinions, measurements to survey responses, and environments to different countries or states. Our model infers meaningful high-level concepts and plausible causal relations among them, demonstrating its utility for learning causal representations of complex real-world data.

Textual event records, such as alarm logs, have become an increasingly common data source in engineering and manufacturing systems. Beyond identifying correlations or recurring patterns, engineers are often interested in understanding which types of events causally trigger or influence other events during system operation. Textual event descriptions may contain semantic clues about such causal relationships, and recent large language models (LLMs) provide a promising tool for extracting these signals. However, relying solely on LLM-encoded textual information is insufficient for accurate causal discovery, since semantic patterns do not directly reveal causal mechanisms and may confuse causation with correlation or frequent sequential patterns. To address these challenges, we propose \textbf{LMT}, a Bayesian causal discovery framework for engineering event data that jointly leverages textual descriptions and timestamps. Specifically, LMT first uses LLMs to extract semantic causal signals from event descriptions and constructs a prior distribution over causal graphs among event types or event clusters. It then incorporates temporal evidence through a Poisson-process-based likelihood, allowing the LLM-informed prior to be refined by timestamp-based statistical evidence. By integrating the textual and temporal information, LMT produces a causal graph that is both interpretable and data-supported. Simulation studies show that the proposed framework is effective across different settings and is especially advantageous in small-sample alarm-event scenarios.

We propose an extended family of structured spatial priors that incorporates the total variation (TV) function with $\ell_p$ norms. The prior is proven to be proper and incorporated into a Bayesian regression framework to enable uncertainty quantification in $T_1$ mapping, with posterior inference performed using the No-U-Turn Sampler (NUTS). This TV--$\ell_p$ construction is proven to constitute a well-defined family of prior distributions, and it naturally enforces spatial consistency and smooth variations in the estimated parameter maps. The method was evaluated in comparison to maximum-likelihood estimation and several Bayesian alternative priors based on the uniform, Gamma, and bounded TV priors. The evaluation includes experiments on synthetic brain and cardiac $T_1$ mapping datasets, as well as a real in-vivo breast $T_1$ mapping dataset. The results show that the TV--$\ell_p$ prior yields more concentrated posterior densities, indicating reduced uncertainty. It also consistently achieves lower variance and smaller (negative) bias, leading to more reliable estimates. Overall, embedding a TV-based structured penalty along with $\ell_p$ norms in a prior in a Bayesian model improves spatial coherence in $T_1$ maps and enhances uncertainty quantification, offering a robust approach for $T_1$ mapping with uncertainties.

We consider multi-environment prediction problems. We assume the environments change the distribution of a latent variable, while the mechanisms generating observed covariates and targets remain stable conditional on that variable. For example, hospitals or clinical cohorts may differ in the prevalence of latent patient states, even though the relationships between those states, physiological measurements, and outcomes remain unchanged. Given a dataset from multiple environments, we formulate a Bayesian model for such problems and derive the corresponding variational objective. We show that this objective decomposes into per-environment terms and an additional cross-environment balancing term induced by the model's structure. We use an empirical Bayes method to set the prior and incorporate it into the objective. Based on this objective, we develop an amortized variational algorithm for posterior approximation, and use the resulting learned latent variables to form predictions in new environments.We study our approach through simulations and real-world studies of astronomical source identification, microbiome-based disease detection, and ICU sepsis prediction. Across these settings, our method outperforms previous approaches for prediction in new environments.

Marginal inference in discrete graphical models forces a choice between exactness and scalability: exact algorithms are intractable for high-treewidth graphs, while iterative approximations (Belief Propagation, variational methods) sacrifice convergence guarantees on frustrated topologies. We argue that this dichotomy stems from a mismatched inductive bias: iterative methods abandon the sequential elimination structure that makes exact inference correct. We introduce In-Context Graphical Inference (ICG-I), an autoregressive Graph Transformer that restores this structure by mimicking Variable Elimination with learned, Tensor- Train-compressed intermediate factors, paired with a Dirichlet output layer and Weighted Conformal Prediction for calibrated, distribution-free coverage guarantees under topological shift. We prove that TT compression errors propagate at most lincarly through the autoregressive chain, that the Dirichlet-Multinomial loss is a proper scoring rule, and that WCP maintains coverage with a quantifiable degradation under estimated density ratios. We conducted intensive experiments to evaluate ICG-I and achieved state-of-the-art performance across all benchmarks. ICG-I reduces MAE from 0.041 (best baseline) to 0.020 on standard instances and achieves 0.048 on N=500 frustrated spin glasses where BP diverges entirely.

Sequential decision-making problems are often modelled as a Markov decision process (MDP). We focus on the stochastic shortest path (SSP) problem, which is an infinite-horizon undiscounted MDP with absorbing terminal states. We develop a Bayesian framework to learn the optimal decision strategy through interactions with the decision-making task. Specifically, we learn the optimal action-value function $Q^*$, but unlike many existing Bayesian approaches, we do not rely on unrealistic modelling assumptions and ad-hoc approximations. Our approach is to directly construct the posterior beliefs for $Q^*$ through Bellman's optimality equations. For deterministic rewards, we characterise the posterior as a distribution with a manifold density. To facilitate simpler inference, we relax the likelihood so that a Lebesgue density exists. The flip side is to create unidentifiability issues. Specifically, the relaxed posterior can have significant mass on improper decision rules, while the exact posterior will not. We also calculate the exact posterior probabilities for optimal action selections for the tabular parametrisation of $Q^*$, a Gaussian likelihood relaxation and a Gaussian prior, which is useful in benchmarking studies. Numerical studies on variants of the Deep Sea benchmark verify our findings. We demonstrate that our framework faithfully quantifies uncertainty and, compared to other temporal-difference-based Bayesian methodologies, is more data efficient. We conclude with recommendations for future work.

Sensor-based Human Activity Recognition (HAR) models often degrade on unseen users due to domain shifts caused by individual movement patterns and sensor placement. Practical wearable HAR systems therefore require personalization methods that are lightweight, applicable across diverse calibration scenarios, and robust under limited calibration data. We present BayaHAR, a gradient-free framework that repurposes pretrained HAR classifiers as Prototypical Networks using prior prototypes that preserve zero-shot performance while regularizing adaptation. For labeled calibration data, we introduce closed-form Bayesian prototype estimation and extend the same principle to weakly labeled data, requiring only knowledge of which activities were performed. With only 3 seconds of calibration (one shot) per activity, supervised adaptation improves test macro-F1 on unseen users by +2.76 to +33.44 percentage points across four datasets, while weakly supervised adaptation improves by +0.56 to +32.13 points. Since adaptation requires only closed-form prototype updates, the framework enables efficient and robust on-device personalization of preexisting HAR classifiers.

One of the primary challenges in Bayesian inference on the parameters of a diffusion model from discrete observations is the unavailability of an analytical expression for the transition density function between consecutive observation times, which is needed to derive the likelihood function. Extending previous studies that solve Fokker-Planck (FP) type partial differential equations with Normalizing Flows, we propose a new Normalizing Flow architecture to learn the transition density function of the diffusion process between two observation times. We do so by solving in a Neural Galerkin framework the associated FP equation with a Dirac mass as initial condition, over a specified training distribution of the initial datum and the coefficients of the diffusion. We specifically focus on processes whose diffusion matrix vanishes in certain inaccessible boundary regions, such as Stochastic Volatility models that satisfy a Feller condition. The product of the obtained transition densities evaluated along the observed trajectory approximates the likelihood function, thereby enabling cheap posterior sampling via Markov chain Monte Carlo (MCMC). After the offline training phase, inference becomes significantly more efficient, as it avoids the need to solve the FP equation in real time for each parameter proposed by the MCMC sampler or to rely on other likelihood-free methods for Bayesian inference that involve repeated simulation of diffusion bridges.

Physics models are inherently imperfect due to misspecified or missing mechanisms, resulting in systematic discrepancies between model predictions and real-world observations. The Kennedy-O'Hagan (KOH) framework addresses this issue through explicit discrepancy modeling. However, its non-amortized, per-instance formulation limits scalability across families of related systems. We introduce Amortized Physics-Informed Calibration (APIC), a population-level extension of KOH that leverages Neural Processes to perform scalable Bayesian inference across realizations. Our framework employs a two-branch latent architecture to disentangle instance-specific physical parameters from shared, state-dependent structural discrepancies. By integrating differentiable physics into an amortized inference backbone, APIC enables rapid calibration of unseen realizations from sparse observations while quantifying uncertainty. Experiments on the damped spring oscillator, the Lotka-Volterra system, and the advection-diffusion PDE with misspecified physics demonstrate improved parameter recovery and consistent identification of the systemic discrepancy structure compared to other calibration approaches.

Representation learning is often described as preserving the information in an input that is relevant for prediction. This work asks what relevance means for a fixed supervised decision problem. A representation is defined to be Bayes-sufficient for a joint distribution and loss if some prediction head can use it to implement a Bayes-optimal action rule. This makes the target information loss-dependent. In the almost-surely unique Bayes-action case, the relevant object is a Bayes quotient, which identifies inputs that require the same Bayes-optimal action. A representation is sufficient when it refines this quotient, and Bayes-minimal when it is informationally equivalent to it. The framework connects naturally to property elicitation: zero-one loss requires the Bayes class, squared loss the conditional mean, Brier loss the conditional probability in binary prediction, and log loss or strictly proper scoring rules the predictive distribution. Controlled finite experiments, learned neural bottleneck experiments, and a real-data iNaturalist taxonomic refinement experiment illustrate the distinction between sufficiency, minimality, and retained non-required information. For a fixed supervised problem, the distribution and the loss determine the Bayes action, the Bayes action determines the quotient, and the quotient determines the minimal information required for Bayes-optimal prediction.

Low-rank tensor decomposition (TD) is usually effective on clean, fully observed data, but it often degrades under severe missingness or noise. Low-rankness is itself a useful but limited structural prior, and additional handcrafted priors (e.g., sparsity or smoothness) still fall short of capturing the rich statistics of real-world data. To compensate for this weak inductive bias under heavy corruption, one would like to inject a learned, data-driven prior; however, the state-of-the-art diffusion models are not readily compatible with current TD and tractable posterior inference. To address these challenges, we introduce DiffBCP, a hybrid-prior Bayesian CP decomposition framework that couples a cumulative shrinkage process prior over the CP factors for automatic rank selection with an off-the-shelf pre-trained diffusion model as an implicit data prior on the reconstructed tensor. To make posterior inference tractable despite the coupling among the likelihood, low-rank constraint, and diffusion prior, we develop a split Gibbs sampler: CP factors admit conjugate updates, while the diffusion block is sampled via low-rank-guided denoising. A noise-adaptive coupling schedule further reduces sensitivity to hand-tuned annealing. Experiments on image inpainting and denoising, including high-resolution out-of-distribution images, show consistent gains over Bayesian, nonlinear, and plug-and-play TD baselines.

Bayesian optimization (BO) is a popular and effective approach for tuning expensive, noisy experiments, but requires the formulation of an explicit objective function. Preferential BO (PBO) removes this requirement by learning from pairwise human feedback, yet existing methods struggle to efficiently optimize beyond low- and medium-dimensional problems due to their global search approaches. We address this limitation by developing a family of local PBO methods that transfer key ideas from high-dimensional BO to the preferential setting. In particular, we introduce local PBO methods which adapt trust-region and derivative-informed local search to pairwise preference feedback, where the latter exploits first- and second-order derivatives of the Laplace-approximated GP posterior. Our benchmark on GP sample paths, standard optimization benchmark functions, and policy-search tasks shows that local PBO methods are especially effective in high-dimensional and complex landscapes with steep optima. Compared with global preference-based baselines, they can substantially reduce cumulative regret, making them particularly useful for real-world preference-based optimization tasks such as policy search.

Shapley values are a principled attribution measure widely used in interpretable machine learning, but their exact computation scales exponentially with the number of players, motivating a wide range of approximation methods based on value function evaluations of sampled coalitions. This raises the question of whether approximation accuracy can be improved by adaptively selecting coalitions for evaluation based on previous evaluations. This is particularly relevant in settings where the value function is costly and the number of evaluations is severely limited, such as retraining-based feature importance, data valuation, and hyperparameter importance. For this purpose, we propose ShaplEIG, a Bayesian experimental design approach that approximates the expensive value function using a Gaussian process surrogate and adaptively selects coalitions based on their expected information gain about the Shapley values. By the linearity of the Shapley values in the value function, we show that the expected information gain is available in closed form. Furthermore, we propose an efficient computation scheme that reduces the complexity from exponential to polynomial in the number of players via elementary symmetric polynomials. In extensive experiments across diverse costly applications, our method consistently improves sample efficiency in the low-budget regime over state-of-the-art baselines.

Predicting whether an individual with Alzheimer's disease will experience mild or severe disease progression is essential for personalized treatment. Typically, practitioners seek to predict the distribution of a discrete disease score, conditional on an individual's current MRI volume and their historical disease trajectory. Classical statistical regression models and single-task neural networks are not well-suited for this purpose because fitting separate models is infeasible (since each individual typically has few observations), while ignoring individual-level correlation leads to poor generalization. Meta-learning, in contrast, provides a natural avenue to dynamically predict distributions without retraining and model nonlinear relationships between the outcome and covariates. Motivated by this, we propose a Bayesian meta-learner that is trained on multiple individuals but tailors the predictive disease score distribution to each individual's historical data. Our model predicts on unseen individuals without retraining, scales linearly with the number of historical observations, and is guaranteed to be less overconfident when predicting long-term disease scores compared to its deterministic counterpart. On real-world data from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database, our model achieves performance competitive with both single-task models and deterministic meta-learners, while substantially improving performance when predicting long-term disease progression.

Implicit-process priors define distributions over functions through flexible generative mechanisms, making them attractive for Bayesian function-space modelling. However, performing posterior inference with such priors is challenging because their induced function-space distributions are typically not available in closed form. One practical strategy is to approximate the prior using a finite collection of sampled functions, and then represent posterior functions as learned combinations of these samples. Existing approaches commonly place a Gaussian variational distribution over the combination weights. While tractable, this choice limits the shapes of posterior uncertainty that can be represented, especially when the true posterior is asymmetric, heavy-tailed, or multimodal. We propose Flow-Transformed Implicit Processes (FTIP), a variational inference method that makes this finite-dimensional function-space approximation more expressive. Instead of using a Gaussian distribution over the combination weights, FTIP uses a normalizing flow to define a richer variational distribution. This induces a flexible posterior distribution over functions while preserving tractable optimization. We train the model using a Black-Box α objective, allowing us to compare mass-covering and mode-seeking variational behaviour. Experiments show that FTIP captures asymmetric and multimodal posterior structure in function space that Gaussian coefficient approximations tend to smooth or collapse.

The Bayesian paradigm offers principled tools for sequential decision-making under uncertainty, but its reliance on a probabilistic model for all parameters can hinder the incorporation of complex structural constraints. We introduce a minimalist Bayesian framework that places a prior only on the location of the optimum, while eliminating nuisance parameters through profile likelihood. This yields a generalized posterior that naturally accommodates structural constraints. As a direct instantiation, we develop MINimalist Thompson Sampling (MINTS). For multi-armed bandits with mean constraints, we establish near-optimal non-asymptotic regret guarantees and sharp almost-sure asymptotic regret characterizations. In particular, MINTS attains the classical Lai--Robbins constant in the unstructured setting and automatically adapts to unimodal structure, achieving the sharp constant determined only by the immediate neighbors of the optimal arm.

Bayesian inference provides a principled framework for modeling epistemic uncertainty in neural networks by treating predictions as distributions rather than deterministic values. Meanwhile, diffusion-based models for 3D molecular graph generation operate on fragile geometric structures governed by strict chemical constraints, making inference highly sensitive to uncertainty miscalibration. A largely overlooked issue is that epistemic uncertainty arising from the learned denoiser interacts with the aleatoric uncertainty intentionally injected during reverse diffusion, leading to systematic variance inflation and a mismatch between the true distribution and the simulated distribution. This effect is particularly detrimental for high-precision molecular generation, where even small deviations can violate chemical validity. In this work, we provide a theoretical and empirical analysis of how epistemic uncertainty propagates through diffusion inference and degrades sampling quality. Building on this investigation, we propose UCD (Uncertainty-Calibrated Diffusion), a simple yet effective method that calibrates the reverse diffusion process to account for epistemic uncertainty. Extensive experiments on standard 3D molecular benchmarks demonstrate that UCD consistently improves sampling quality across diverse baseline methods, establishing new state-of-the-art performance for 3D molecular diffusion. The code is available at https://github.com/jiuguaiwf/UCD.

We ask: when do Bayesian model averaging (BMA) weights over decision trees carry sufficient epistemic information to justify committed exploitation of the averaging distribution? We answer this question in closed form for Bayesian decision trees (BDTs) with Dirichlet-Multinomial leaf models and a Catalan-exponential tree-size prior (Schetinin&Jakaite, 2025), establishing a complete non-asymptotic theory of rational commitment thresholds.

The quadratic complexity of self-attention remains a bottleneck for Large Language Models (LLMs) processing ultra-long contexts. The Naive Bayes Cognitive Engine (NBCE) parallelizes long-context inference by chunking documents and routing to the lowest-entropy chunk at each decoding step. This hard-selection strategy causes semantic fragmentation during cross-chunk reasoning, as abrupt routing changes between adjacent tokens disrupt the model's contextual grounding. We present Soft-NBCE, a lightweight extension that replaces discrete chunk selection with soft entropy-weighted chunk fusion. A temperature-scaled Softmax over predictive entropies assigns continuous weights to all chunks, enabling log-space aggregation across chunk-conditioned distributions. To partially compensate for the conditional independence assumption introduced by chunking, we propose Consistency Distillation, a LoRA-based self-distillation that constrains the chunked logit distribution toward a full-context teacher via KL-divergence. On LongBench multi-hop benchmarks, Soft-NBCE with Consistency Distillation improves consistently over NBCE-style baselines (MuSiQue F1: 0.310 vs.\ 0.275 for Vanilla NBCE; HotpotQA F1: 0.479 vs.\ 0.427) while maintaining retrieval accuracy (NIAH-32K: 0.909) at O(L^2/n) peak memory.

Editing complex, long-form knowledge in Large Language Models remains a significant challenge due to the difficulty of maintaining generation coherence. Existing autoregressive methods like AnyEdit alleviate length constraints but rely on Fixed-window Chunking, which disregards logical structure and compromises consistency. To address this, we present AnyEdit++, a structure-aware framework incorporating Bayes-Chunk, an adaptive segmentation mechanism that dynamically identifies semantic boundaries based on Bayesian Surprise. We underpin this approach with a theoretical framework establishing two key principles: (1) Structural Independence: we prove that cross-segment interference is minimized when anchor keys are geometrically orthogonal (a condition naturally satisfied by our surprisal-based boundaries but violated by fixed windows), and (2) Causal Locality: we demonstrate that updates injected at these semantic peaks yield strictly superior control compared to arbitrary split points. Extensive experiments across mathematical reasoning, code generation, and narrative tasks demonstrate that AnyEdit++ achieves superior performance and robustness compared to state-of-the-art baselines, validating that structural awareness is critical for effective long-form knowledge editing.

This paper studies a one-step lookahead Bayesian optimization (BO) method and its theoretical guarantee. Although the empirical effectiveness of one-step lookahead BO methods, such as entropy search, has been studied extensively, they often rely on computationally intractable approximations, and their regret guarantees remain underdeveloped. Thus, this paper proposes a one-step lookahead BO method called optimal-point variance reduction (OVR), which requires only posterior sampling and Monte Carlo approximations. We obtain a uniform error bound over an input domain for the Monte Carlo estimation in OVR. Furthermore, we show that the regularized OVR, with the slight modification to promote exploration, achieves a vanishing Bayesian expected simple regret upper bound. Finally, we demonstrate the effectiveness of OVR through numerical experiments.

Multi-arm bandit algorithms are increasingly used in online platforms, clinical trials, and social science experiments, but valid statistical inference on their performance remains an open challenge. After deploying bandits, a natural question is whether one can construct a confidence interval for its mean reward and assess whether it reliably outperforms a baseline policy. The total reward achieved in any single bandit deployment is random, and deploying a bandit twice on the same population typically yields different reward trajectories due to stochastic rewards. Standard statistical inference methods cannot be used because bandit algorithms introduce complex dependencies in the collected data, which violate the i.i.d. assumption underlying many classical approaches. Moreover, existing inference methods for adaptively collected data only apply to estimands that do not depend on the data-collection algorithm (such as the mean reward under a fixed action). We propose Bandit Simulation for Inference (BSI), a framework that fits a simulator of the bandit environment from observed data--either on-policy or off-policy--and uses it to estimate the mean reward under any evaluation policy, including adaptive blackbox algorithms. BSI formally propagates uncertainty in the estimated simulator parameters into the confidence interval construction. Furthermore, for BSI to be valid, it requires only weak exploration assumptions on the behavior policy and avoids importance weighting. We prove that BSI yields asymptotically valid confidence intervals, and demonstrate empirically that it maintains nominal coverage in settings where standard off-policy evaluation methods fail.

Reliable quantification of malaria dynamics in sub-Saharan Africa is hindered by short, noisy, and spatially heterogeneous surveillance records. In Ghana, health-facility data from 2014 to 2023 reveal non-linear and age-specific fluctuations in hospital admissions, yet existing approaches struggle to capture stochastic variability or provide credible uncertainty bounds. This study develops a Bayesian nonlinear inference framework that integrates a cubic baseline with a damped oscillatory kernel, estimated via an affine-invariant ensemble Markov Chain Monte Carlo sampler. The framework accommodates limited data, models parameter uncertainty, and generates probabilistic forecasts for children under five years and individuals aged five years or more. Results show strong empirical adequacy ($R^2 = 0.9958$ for $<5$ years; $R^2 = 0.9956$ for $\geq 5$ years) with residual errors below $2\%$ and well-mixed posteriors confirming convergence. District-level analysis reveals pronounced spatial heterogeneity, with coefficients of variation ranging from $<0.07$ in urban centres such as Kumasi to $>3.3$ in peripheral districts such as Mpohor and Bia East. Forecasts for 2024-2026 indicate a gradual resurgence: from 137,000 to 149,000 cases among children under five years and from 348,000 to 375,000 cases among older individuals, with uncertainty widening over time. By producing probabilistic forecasts, this Bayesian framework provides a principled tool for anticipating malaria fluctuations and strengthening data-driven decision-making in Ghana's national malaria control strategy.

Genotype-based cis-expression prediction depends on accurately modeling local regulatory architecture. We present block-sparse Bayesian sparse linear mixed model (bsBSLMM), an extension of Bayesian sparse linear mixed model (BSLMM) that incorporates linkage disequilibrium (LD)-block spike-and-slab sparsity and a transcription start site (TSS)-informed SNP inclusion prior. Across 23,098 genes from GEUVADIS European-ancestry lymphoblastoid cell lines, bsBSLMM retained more predictable genes than BSLMM, LASSO, BLUP, TIGAR elastic net, and TIGAR Dirichlet-process regression under matched evaluation criteria. Compared with BSLMM, bsBSLMM improved held-out prediction performance for most shared genes, with gains driven primarily by LD-block sparsity and further enhanced by the TSS-informed prior. Variants selected by bsBSLMM showed stronger enrichment in GM12878 DNase and H3K27ac regulatory regions than variants selected by BSLMM. In transcriptome-wide association study (TWAS) analysis, bsBSLMM recovered established inflammatory bowel disease signals, including IL23R, and identified additional genome-wide significant genes not detected by BSLMM. Independent validation in the Louisiana Osteoporosis Study reproduced the increased prediction yield across ancestries and recovered biologically relevant bone mineral density pathways in downstream TWAS and gene set enrichment analyses. These results demonstrate that incorporating LD-block structure and biologically informed SNP priors improves cis-expression prediction and enhances downstream TWAS discovery.

When many reinforcement-learning policies achieve near-optimal return, a post-hoc auditor may have to distinguish among many behaviorally distinct but return-equivalent policies. We formalize this phenomenon through an occupancy-measure analogue of Rashomon capacity: the metric entropy of the near-optimal occupancy region, computed relative to an audited deployment class. Because occupancy measures identify behavior only up to occupancy equivalence, we formulate auditing at the occupancy-class level and distinguish exact local-query oracles from noisy sample-query oracles. Our main exact-query result is conditional: if the audited class contains a $2/H$-separated near-optimal packing whose local signatures are $b$-sparse, then exact local-query auditing requires $Ω(M/b)$ queries; when the packing realizes deployment-class capacity and $b=O(1)$, this becomes $Ω(2^{\Hopt^\cF(\eps)})$. We give a finite discounted hidden-branch MDP attaining this bound and show the exact Bayes success law. For noisy hidden-trigger testing, we prove a mixture lower bound of order $M/β$, where $β$ is the per-sample KL signal, yielding $Ω(2^{\Hopt^\cF(\eps)}/(ρ^2Δ^2))$ for capacity-order packings with $β=O(ρ^2Δ^2)$. We also provide a static target-recognition information lower bound, a transcript-compatible oracle-cover verification upper bound, and a canonical occupancy regularizer whose regularized audited capacity collapses when a trusted reference occupancy is available. Controlled benchmarks distinguish positive sparse-signature instances from high-capacity negative controls where exact auditing is easy, and map the noisy-trigger law to post-processed continuous-control and visual-RL auditing regimes.

Financial retrieval-augmented generation (RAG) systems typically rank evidence by textual relevance, but in financial markets evidence utility depends on event type, forecast horizon, and market context. We study news-triggered event-impact prediction as a point-in-time financial RAG problem. For each company-news anchor, the system retrieves financial news and SEC filing passages, appends a pre-decision market-context card, and predicts multi-horizon residual-return signals. Our method keeps the LLM frozen and adapts retrieval through an external Bayesian source memory updated from matured residual-return feedback. On a fixed 89-stock Nasdaq-oriented universe derived from the FinRL-DeepSeek/FNSPID task, using original FNSPID news and point-in-time EDGAR filing passages, Frozen Reader with Source Memory improves held-out macro-F1 from 0.438 to 0.471 and downstream portfolio Sharpe from 0.52 to 0.84 relative to Frozen Reader with No Memory. Supervised LoRA gives modest gains under static retrieval, but after source-memory adaptation, the LoRA reader does not improve over the frozen reader. These results suggest that, for financial RAG systems, learning where to retrieve can be as important as learning how to read, offering a modular route to market-feedback adaptation.

We introduce the design-model framework: a way to derive efficient recurrent sequence maps from explicit assumptions about memory. A design model writes evidence into memory by exact Bayesian filtering; a query-dependent readout produces a predictive distribution whose mean is the layer output. In our linear-Gaussian instantiation, the \emph{Bayesian Layer} propagates both a mean and a covariance: the covariance tracks uncertainty over stored associations, steering writes toward uncertain directions, attenuating gains as evidence accumulates, and preserving confident memories. The same framework unifies several sub-quadratic recurrences. Linear attention, GLA, and Mamba-2/SSD are exact filters under one design model, whereas DeltaNet and related Delta-rule models arise as covariance-reset reductions under another. Restoring the covariance yields closed-form predictions for retrieval dynamics, verified empirically, and improves robustness beyond the training regime across controlled collision studies, learned associative recall, and the Zoology MQAR benchmark; distilling Bayesian Layers into a pretrained 340M Gated DeltaNet improves RULER long-context retrieval at matched compute.

Nonlinear conservation laws are at the heart of many of the most important dynamical systems in science and engineering. In practical applications, such systems are often subject to various sources of uncertainty, e.g. due to sparse or noisy measurements. Inferring physical quantities and fields of interest then becomes an ill-posed problem which both classical numerical methods and modern deep learning-based methods struggle to treat appropriately. Recent work has framed classical numerical methods as Bayesian inference under Gaussian process priors, resulting in a physics-aware treatment of uncertainties. Following this line of work, we develop a novel numerically conservative method for uncertainty-aware simulations of nonlinear conservation laws. We use recent sparse approximation techniques to scale up to large-scale forward and inverse problems. For forward simulation, we inherit the accuracy of classical solvers while providing structured uncertainty quantification. On inverse problems, we recover posteriors over nonparametric source fields in seconds -- outperforming neural baselines that take minutes to produce a less accurate point estimate.

A central aim of deep learning theory is to characterize how neural networks make predictions in the regime of simultaneously large model and training set size. Since the limits of diverging number of model parameters and dataset size do not commute it is not clear a priori what limits exist. In this work, we shed new light on these questions by studying Bayesian inference in deep non-linear MLPs in the regime where the number of training samples ($P$), the input dimension ($N_0$), the hidden layer width ($N$), and the number of hidden layers ($L$) can all be large. We build on the Neural Covariance SDE (Li et al., 2022) to analyze predictive posteriors in the regime where $LP/N\inΘ(1)$, playing the role of an effective network depth. Our framework covers both smooth and ReLU activation functions and applies to arbitrary temperature. We find to first order in $LP/N$ a simple criterion for which data generating processes benefit from depth in the sense that larger $LP/N$ increases the Bayesian model evidence. We also give a novel derivation of a prior result from the physics literature that at least to first order in $LP/N$, the Bayesian predictive posterior is remarkably simple and is simply equivalent to that of a data-dependent kernel method.

Graph classification is a core task in graph data mining with widespread real-world applications. Recent advances in graph neural networks (GNNs) have led to substantial performance improvements for graph classification. However, existing GNNs are typically forced to make predictions even under high uncertainty or unknown conditions, resulting in unreliable decisions that can severely impact downstream tasks, particularly in safety-critical scenarios. To address this critical limitation, we propose AbstainGNN, a novel and theory-driven framework for graph classification with abstention, which enables GNNs to reject uncertain predictions instead of producing incorrect decisions. Specifically, AbstainGNN explicitly models both the predictive function and the abstention function, allowing for effective utilization of graph structural information. Moreover, unlike existing heuristic abstention methods, we theoretically characterize the trade-off between classification errors and rejection costs from a PAC-Bayesian generalization perspective, and derive a unified learning objective for model optimization. Guided by this theoretical insight, we further develop an efficient two-stage training strategy consisting of predictive function warm-start and abstention function calibration. Extensive experiments on five benchmark datasets show that AbstainGNN outperforms existing abstention methods, achieving superior classification performance under the same rejection rates.

Parameter-efficient adaptation has made LLMs practical for domain prediction, but standard LoRA still relies on a static low-rank update and does not expose the latent interactions that often drive scientific labels. We introduce iLoRA. To our knowledge, it is the first Bayesian graph-conditioned LoRA framework. It infers a latent interaction graph from the input and uses it to generate input-conditioned LoRA updates. As a result, iLoRA learns prediction and latent interaction structure jointly, rather than training a predictor and applying interaction analysis only post hoc. We instantiate this idea for microbiome diagnosis, where disease state can depend on both species-level abundance and microbe-microbe cross-talk, and evaluate it in two complementary settings: interactive QA with human-annotated graphs, which tests latent structure recovery, and multi-cohort IBD diagnosis, which tests biomedical utility. Across both settings, iLoRA improves over strong LoRA and Bayesian adaptation baselines, recovers graphs aligned with human annotations and cohort-level microbiome associations, and provides calibrated uncertainty with moderate graph-branch overhead.

Sparse recovery in linear systems underpins applications from signal processing to high-dimensional regression. Sparse Bayesian Learning, grounded in the principle of automatic relevance determination (ARD), offers a practical Bayesian mechanism for feature sparsity via marginal likelihood optimization. Yet, its reliance on a homoscedastic noise model renders it sensitive to data contaminations such as outliers or misspecified noise, harming model fit and predictions. Instead, we propose jointly learning individual feature and sample relevancies, enabling simultaneous model and data sparsification via a single Bayesian objective. This symmetric pruning of model and data offers a natural extension that preserves conjugacy, admits closed-form updates for standard optimization procedures, and aligns with perspectives from robust regression and influence functions. Empirical results across diverse regression tasks affirm that a joint ARD approach consistently yields both sparse and robust prediction models.

Critical sequential decisions are rarely single-timescale: a strategic decision causally shapes the context in which every subsequent tactical choice is made; standard bandit and reinforcement-learning theory does not capture this causal coupling between timescales. We formalise the problem class as Nested Contextual Causal Bandits (NCCBs), a hierarchical SCM where each level's action sets the next level's context distribution, and propose Nested Causal Thompson Sampling (NCTS), which draws one mechanism-factorised belief per episode and acts recursively under it. Our main theoretical result is a causal PAC-Bayesian excess-risk bound that certifies any candidate deployment policy from historic data alone, off-policy and anytime, answering the deployment question: can we trust this agent here, and at what risk? Experiments on a hierarchical SCM show that, against a matched RFF-GP joint regression on the same function class, the factorised SCM-mechanism posterior transfers significantly better zero-shot under exogenous distribution shifts, the recursive meta-to-inner commit significantly dominates the joint-commit alternative in distribution, and the certificate significantly contracts as offline data accumulates. Combining these results, we establish progressive certified handover, a safe-deployment method: each timescale flips from a legacy controller to NCTS when gains can be certified, independently of the others.

The scaling limit where both the size of the training set $P$ and the width $N$ of a deep neural network grow at the same rate, the so-called proportional-width regime, has been intensely studied for shallow, single-hidden-layer networks. However, extending these non-perturbative results from shallow architectures to deep non-linear networks has proven very challenging. Here we present an effective approximate approach to predict the generalization performance of Bayesian multi-layer perceptrons (MLPs) of fixed depth $L$ on arbitrary high-dimensional data. We propose an equivalent Wishart Ansatz to capture the dominant stochastic fluctuations of the hierarchical empirical kernels of MLPs. This allows us to perform a large deviation analysis for the partition function of MLPs in the proportional limit, expressed in terms of a renormalized NNGP kernel. In this description, even strong representation learning in the proportional limit is encoded in at most $L$ scalar order parameters, determined self-consistently. Extending the approach to convolutional architectures (CNNs), we identify a hierarchical local kernel renormalization mechanism, which allows to quantify more complex data-dependent transformations of the large-width kernel in CNNs due to finite-width effects. We test our effective theory against sampling experiments from the Bayesian posterior of finite deep neural networks with depths $L \sim O(10)$ and $P\sim O(10^3)$ on classic benchmark datasets, finding overall very good agreement together with two distinct types of systematic deviations.

While parameter-efficient fine-tuning methods like low-rank adaptation (LoRA) are standard for large language models, principled estimation of epistemic uncertainty remains challenging. Recent results in the LoRA regime suggest that discrete multi-mode approaches such as deep ensembles offer little benefit over single-mode methods. This contradicts broader observations in deep learning, where ensembling independent optima typically improves generalization, and linking these modes through continuous low-loss valleys further enhances Bayesian model averaging (BMA). Whether such structure exists in the LoRA space and whether it yields functional diversity missed by local or discrete methods has not been studied. We introduce LoRA-Curve, a segmented Bézier curve parameterization in the LoRA space, with two variants: a free configuration that jointly optimizes all control points, and an anchored configuration that connects independently fine-tuned LoRA optima. We prove pathwise continuity and Lipschitz regularity of the loss along the curve and empirically show, across reasoning and classification benchmarks with Qwen2.5 7B, that linear interpolation encounters loss barriers, while our anchored multi-segment curves connect independent optima through continuous low-loss valleys. Combined with flat-minima perturbations and a Jensen-Shannon divergence regularizer, LoRA-Curve yields measurably higher mutual information of the predictive distribution without sacrificing performance, and links continuous parameter-space traversal to functional diversity.

Solving high-dimensional PDE-governed inverse problems is often challenging due to complex non-Gaussian posterior distributions, expensive forward model evaluations, and misspecified prior information. To address these issues, we propose a deep adaptive dimension-reduction Bayesian inference framework based on the Variational Flow (VF) model. Since standard normalizing flows are restricted by bijective mappings and cannot directly reduce dimensions, VF overcomes this limitation by integrating VAE-based nonlinear dimension reduction with dual normalizing flows for the latent prior and encoder. This design provides a strictly higher evidence lower bound than VAE and allows more flexible approximation of complex posterior distributions. We further introduce an iterative prior updating strategy that gradually moves the prior mean toward high-probability posterior regions, avoiding manual prior tuning. These components form a closed adaptive loop together with an adaptively fine-tuned Fourier Neural Operator (FNO) surrogate: VF generates posterior-concentrated samples to refine the surrogate, while the updated surrogate further improves posterior inference. Numerical experiments on a 100-dimensional Rosenbrock problem and three standard PDE-governed inverse problems show that our method delivers competitive or superior accuracy compared with MCMC, UKI, and SVGD baselines across all tested configurations, with the most pronounced advantages emerging in challenging scenarios such as high-noise observations and high-dimensional parameter spaces.

We study minimal attention-only transformers under all-token corruption and show they admit a two-stage empirical Bayes interpretation. A single attention step computes a kernel-weighted posterior mean with respect to the empirical distribution defined by the context. Depth refines this distribution through particle dynamics (Stage 1), while a long-range skip-connection carries the noisy input as a query for posterior inference (Stage 2), revealing distinct statistical roles for depth and attention residuals. The framework isolates a minimal setting in which the context itself induces a depth-dependent energy landscape governing in-context inference. We show that effective denoising can emerge without an explicit noise schedule: a fixed kernel bandwidth and finite integration horizon suffice, yielding a principled depth-noise relationship. We further establish a posterior-mean recovery guarantee for a class of well-behaved priors, where the empirical estimator converges to the Bayes-optimal predictor under asymptotic conditions. Connecting these dynamics to reverse-diffusion limits, our results provide a statistical interpretation of attention as in-context inference via sample-based posterior estimation, without explicit density modeling.

Improving the quality of model-generated summaries, especially factuality, the accuracy of a summary with respect to its source content, remains a challenge. While reranking could select the optimal output from multiple generated candidates, it is limited to only using the source as guidance, resulting in unreliable summaries. To address this limitation, we propose ConSUM that reranks candidate summaries by considering two factors: consistency to the source document and consensus among the other candidates. Consensus is established using Minimum Bayes Risk (MBR) decoding over the set of generated summaries, while ensuring consistency by employing factuality-aware metrics that compare the summary against the source. Rigorous testing demonstrates that our system is competitive with existing methods, with human evaluations further confirming that its generated summaries are preferred over those from other systems. Our code is available at https://github.com/naist-nlp/ConSUM .

Several important learning tasks can be formulated as minimizing an entropy-regularized objective over an appropriate space of probability distributions. Mean-field Langevin dynamics (MFLD) facilitate computation in this general context, casting the minimizer as the invariant distribution of a McKean--Vlasov process, which can be numerically discretized using $N$ particles and thus simulated. However, simulating this interacting particle system has computational complexity of order $N^2$. Motivated by recent research into \emph{kernel thinning}, we propose \texttt{KT-MFLD}, in which each particle interacts only with a thinned particle coreset of size $\mathcal{O}(N^{\frac{1}{2}})$. \texttt{KT-MFLD} thus reduces the computational complexity to order $N^{\frac{3}{2}}$ while, under mild regularity conditions, achieving the same convergence guarantees (up to logarithmic factors) as MFLD. Our theoretical analysis is empirically confirmed on tasks including the training of student-teacher neural networks, quantization with maximum mean discrepancy, and computation of predictively-oriented posteriors in a post-Bayesian framework.

Simulation-based inference with neural posterior estimation (NPE) often yields overconfident and unreliable posteriors under limited simulation budgets. To address this, we propose DRO-NPE, a distributionally robust approach that replaces the standard NPE objective with a worst-case loss over a Wasserstein ambiguity set. We introduce KL-based metrics for miscoverage and miscalibration, and use these to show that the DRO-NPE objective controls overfitting and reduces posterior overconfidence. Our method is tractable, parallelisable, and readily integrates with standard normalising flows. Across benchmark SBI tasks, DRO-NPE consistently improves coverage and calibration, while narrowing the gap between empirical and population NPE loss, leading to more reliable inference in low-simulation regimes.

Knowledge distillation is a powerful method for model compression, enabling the efficient deployment of complex deep learning models (teachers), including large language models. However, its underlying statistical mechanisms remain unclear, and uncertainty evaluation is often overlooked, especially in real-world scenarios requiring diverse teacher expertise. To address these challenges, we introduce \textit{Multi-Teacher Bayesian Knowledge Distillation} (MT-BKD), where a distilled student model learns from multiple teachers within the Bayesian framework. Our approach leverages Bayesian inference to capture inherent uncertainty in the distillation process. We introduce a teacher-informed prior, integrating external knowledge from teacher models and task-specific training data, offering better generalization, robustness, and scalability. Additionally, an entropy-based weighting mechanism adaptively adjusts each teacher's influence, allowing the student to combine multiple sources of expertise effectively. MT-BKD enhances the interpretability of the student model's learning process, improves predictive accuracy, and provides uncertainty quantification. We validate MT-BKD on both synthetic and real-world tasks, including protein subcellular location prediction and image classification. Our experiments show improved performance and robust uncertainty quantification, highlighting the strengths of our MT-BKD framework.

Causal discovery is a cornerstone of scientific reasoning, yet whether large language models can perform it reliably remains an open question. Recent benchmarks show that even fine-tuned models plateau on simple causal graphs and degrade as complexity grows, but why they fail has not been established. We prove the failure is fundamental: supervised fine-tuning, direct preference optimization, and in-context learning all produce predictors that cannot distinguish between causal graphs generating similar observational data, and any attempt to do so requires the model's internal representations to grow unboundedly, violating the very conditions under which these methods work. We formalize this as a kernel obstruction theorem, establishing that the limitation is intrinsic to the learning paradigm, \emph{not any particular model or dataset}. We propose Agentic Causal Bayesian Optimization (A-CBO), wherein a frozen language model serves as an interventional oracle answering targeted queries about intervention effects, while an external Bayesian loop concentrates beliefs over candidate graphs in logarithmically many rounds. Because the decision operates outside the space where the obstruction applies, A-CBO provably converges while the underlying model remains unchanged. On Corr2Cause, A-CBO matches fine-tuned baselines without any training. On Extended Corr2Cause, a new benchmark scaling to 24 variables with 18K test samples, A-CBO significantly outperforms both fine-tuning and preference optimization, with the advantage growing

Flow and diffusion generative models have established themselves as widely adopted density estimators for simulation-based inference (SBI), extending naturally from neural posterior estimation to likelihood and joint density estimation. Their principled optimization objectives and freedom from architectural constraints have driven rapid adoption across the natural sciences. Yet the most widely used SBI libraries remain PyTorch-based, leaving researchers who develop their forward models and analysis pipelines in JAX without a native option. We present GenSBI, an open-source library that implements flow matching, score matching, and denoising diffusion entirely in JAX. The library offers three transformer-based architectures - SimFormer, Flux1, and a novel Flux1Joint that extends gate-modulated transformer blocks to joint density estimation - all interchangeable through a unified interface that decouples generative method, neural backbone, and inference mode. GenSBI provides an end-to-end workflow from training through posterior calibration (SBC, TARP, LC2ST) and supports custom architectures with domain-specific embedding networks. We validate the framework on standard SBI benchmarks, achieving near-ideal mean C2ST scores (0.50-0.56, where 0.50 is ideal) on SBIBM tasks with minimal per-task tuning and well-calibrated posterior coverage across all tested configurations. The code is publicly available at https://github.com/aurelio-amerio/GenSBI.

Bayesian experimental design (BED) is a principled framework for data-efficient design of sequential experiments. However, existing BED methods are unable to adapt to dynamic constraints inherent in real-world tasks due to budget limitations, varying costs, or physical constraints that restrict how designs evolve over time. In this paper, we introduce a novel approach to BED that enables constrained optimization of experimental designs by combining offline pre-training of an amortized policy and a posterior network with online multi-step lookahead planning using scenario trees. We empirically demonstrate that our method yields substantially more informative design sequences than existing methods across a range of constrained BED tasks, while incurring only a modest additional computational overhead.

Amortized inference promises fast test-time Bayesian inference, but existing methods are inherently tied to fixed models. Extending amortization to unseen models typically requires retraining or costly test-time finetuning. In this paper, we ask: is it possible to build a single inference network capable of generalizing across varying priors, likelihoods, and dimensionality? We introduce Amortized Factor Inference Networks (AFINs), a family of encode-merge-decode inference networks built on dimension-independent modules that map a model specification and its observations to the parameters of a variational posterior. Experimentally, a single trained AFIN achieves posterior accuracy comparable to NUTS and several variational inference methods, while requiring 2 to 4 orders of magnitude less test-time compute. Code is available at https://github.com/joohwanko/AFINs.

Physics-informed machine learning (PIML) integrates mechanistic knowledge, typically in the form of partial differential equations (PDE), into data-driven models. Despite strong empirical performance, its statistical generalisation properties remain poorly understood, particularly in the regression setting with unbounded losses. Existing analyses rely on approximation or stability arguments and do not fully capture how physical structure influences generalisation from finite data. In this work, we develop a PAC-Bayesian framework for PIML that provides high-probability generalisation guarantees in the presence of unbounded losses. We adopt a multi-task perspective that jointly treats data fidelity, PDE residuals, initial and boundary conditions, avoiding the looseness induced by standard union-bound approaches. Our analysis leverages the structure of physics-informed objectives to derive novel bounds where the complexity scales with input-gradient norms of the losses, revealing a direct link between physical regularity and generalisation. We instantiate this framework under Sobolev and Poincaré-type assumptions, yielding two classes of bounds that trade off statistical complexity and smoothness in different regimes. Building on these results, we propose a self-bounding-aware learning algorithm that directly optimises tractable surrogates of the derived bounds, along with a practical procedure to estimate the associated constants in realistic settings. Empirical evaluations on standard PDE benchmarks demonstrate that our bounds are non-vacuous, significantly tighter than union-bound baselines, and can be effectively minimised during training. Overall, our results provide a principled statistical foundation for the generalisation of physics-informed models.