Reinforcement learning from human feedback (RLHF) has emerged as a powerful yet sample-inefficient approach for learning reward models from human preferences, making active learning a critical component in synthesizing informative preference queries. However, effective uncertainty quantification required for active learning remains a key challenge for large neural network reward models. In this paper, we introduce PreferenceEKF, a sample-efficient approach that tracks reward model uncertainty by framing active preference learning as a sequential Bayesian filtering problem. Instead of relying on computationally prohibitive posterior inference over the full neural network parameter space, our method performs sequential inference via an extended Kalman filter within a low-dimensional parameter subspace, continuously updating the reward model posterior as new preference queries arrive. Our approach enables scalable sampling of neural network parameters to efficiently compute acquisition functions for active reward learning. Experiments on the D4RL and V-D4RL benchmarks demonstrate that our approach achieves better sample efficiency, runtime, scalability, and calibration compared to other Bayesian deep learning approaches, and the learned reward models lead to competitive offline reinforcement learning policy performance. This highlights the potential of scalable Bayesian methods for preference-based reward modeling in RLHF. Our code is available at https://github.com/yutaizhou/bnn_pref.
Knowledge graphs describe reality in crisp assertions, while the systems now consuming them, foundation models and autonomous agents, reason natively in probabilities. We argue that this mismatch is why the integration of language models and knowledge graphs remains a data-feeding pipeline rather than a unified reasoning architecture. We envision Semantic Bayesian World Models (SBWMs): a Web that describes the world not as a database of facts but as a shared, evolving fabric of beliefs over knowledge graphs, where ontological axioms constrain priors, observations update beliefs by Bayesian conditioning, and actions intervene upon the world. We work through what an agent gains from such a model: a home-security agent deciding whether the figure at the gate is a courier or a burglar, an actuarial estimate aggregated by entailment rather than by string frequency, a planning task that language models reliably fail, and the estimation of quantities that no document has ever stated. We then set out what the community must build to make them possible: belief annotation over RDF~1.2, probabilistic entailment regimes, semantic calibration layers, and protocols by which agents that have never met can exchange, and disagree over, calibrated beliefs.
This paper addresses infinite-dimensional Bayesian inference for inverse problem of partial differential equations with model parameters in infinite-dimensional Hilbert space. To effectively incorporate prior information, we propose a novel continuous normalizing flows based infinite-dimensional model. Specifically, by introducing a well-defined neural ordinary differential equation in infinite-dimensional space, a simple reference measure can be transformed into a more complex measure which encodes the prior information. A corresponding theoretical framework is established to ensure the well-posedness of our proposed Bayesian prior in infinite-dimensional space. We also provide training methods of the prior for two distinct data settings, along with two sampling algorithms for the resulting Bayesian posterior. The proposed framework is applied to three representative inverse problems: the simple smooth inverse problem, inverse scattering problem, and the inverse heat conduction problem. Numerical experiments support the theoretical analysis and demonstrate the efficiency of the proposed algorithms.
Stochastic gradient Markov chain Monte Carlo (SGMCMC) methods enable scalable Bayesian inference, but their performance depends strongly on hyperparameters such as the step size, mini-batch size, and number of leapfrog steps. Since most SGMCMC algorithms lack a Metropolis-Hastings acceptance rate, standard acceptance-based tuning methods are not directly applicable. We propose HyperMC, a multi-fidelity tuning framework that combines Hyperband-style resource allocation with kernel Stein discrepancy (KSD) evaluation. By running multiple successive-halving brackets, HyperMC balances broad exploration of a continuous hyperparameter space with increasingly accurate evaluation of promising configurations under a fixed computational budget. We further introduce Robust HyperMC, which uses global grid initialization followed by elite-guided local refinement to reduce sensitivity to random candidate generation and noisy finite-budget evaluations. Under suitable approximation and concentration conditions for the estimated KSD, we establish that the successive-halving component selects a near-optimal configuration among the sampled candidates with high probability and derive a sufficient computational budget for successful selection. Experiments on logistic regression, probabilistic matrix factorization, and Bayesian neural networks show that HyperMC improves posterior approximation or predictive calibration relative to MAMBA, grid search, and heuristic baselines, while Robust HyperMC yields more stable and reproducible tuning results.
Toni J. B. Liu, Jiajun Bao, Yizhou Liu +4cs.LG cs.AI cs.CL stat.ML
What does a language model predict when it has few clues? The answer lurks in its unembedding geometry: a single direction of the unembedding matrix encodes the unigram distribution of the training corpus, which serves as the Bayesian prior the model falls back on when uncertain. This structure --- which we term the \emph{direction of ignorance} --- appears in all four model families examined (\texttt{Llama}, \texttt{Qwen}, \texttt{Gemma}, and \texttt{Pythia}), ranging from 0.4B to 405B parameters. Projecting the final prediction state onto this direction yields a per-token \emph{prior loading factor} $λ$, which, empirically, declines steadily as the context becomes more informative. Formally, the same projection decomposes the prediction state into two orthogonal vectors that correspond exactly to the two factors of a tempered Bayesian update: a unigram prior raised to the exponent $λ$ and a context-driven likelihood. This geometric-probabilistic interpretation calibrates $λ$, making it meaningfully comparable across model sizes and families, with larger models generally exhibiting lower prior reliance in the high-context limit. Finally, we show that the direction of ignorance is causally active: raising or lowering $λ$ at the final prediction state steers the prediction toward or away from the unigram prior in KL divergence.
Many statistical models involve parameter-dependent normalizing constants that are computationally intractable, creating substantial obstacles to standard Bayesian inference. Although existing likelihood-based algorithms can often circumvent these constants, their uncertainty quantification may be poorly calibrated under model misspecification. To address these challenges, we propose SME-BETEL, a semiparametric Bayesian framework that combines score matching estimating equations with Bayesian exponentially tilted empirical likelihood. The resulting posterior avoids evaluation of normalizing constants and does not require learning-rate calibration. We establish consistency and asymptotic normality of the score matching estimator, and prove a Bernstein-von Mises theorem for the SME-BETEL posterior. These results show that SME-BETEL credible sets are asymptotically calibrated to the sampling variability of the score matching estimator, yielding valid frequentist coverage under model misspecification. We further develop a new score matching criterion for mixed-domain data, extending SME-BETEL to models whose observations combine components from different sample spaces. This construction enables robust Bayesian inference for mixed-domain doubly-intractable models, including preferential sampling models with an intractable spatial normalizing integral. Simulation studies show that SME-BETEL remains competitive under correct specification and substantially improves uncertainty quantification under misspecification. An ozone-monitoring application further demonstrates the utility of the mixed-domain construction for spatial preferential modeling.
Bayesian methodologies for handling count-valued time series have gained prominence due to their ability to infer interpretable latent structures and to estimate uncertainties. Among these Bayesian models, Poisson-Gamma Dynamical Systems (PGDSs) are proven to be effective in capturing the evolving dynamics underlying observed count sequences. However, the state-of-the-art PGDS still falls short in capturing the transition dynamics that are commonly observed in real-world count time series. To mitigate this limitation, a PGDS with time-varying transition kernel (TV-PGDS), is proposed to allow the underlying transition matrices to evolve over time. Three specifically-designed Dirichlet Markov chains (Dir-Dir, Dir-Gam-Dir, PR-Gam-Dir) are constructed to accommodate heterogeneous structural mutations within these dependencies. Leveraging Dirichlet-Multinomial-Beta data augmentation techniques, a fully-conjugate and efficient Gibbs sampler is developed to perform posterior simulation. Experiments show that, in comparison with related models, the proposed PGDS achieves improved predictive performance due to its capacity to learn time-varying dependency structure captured by the time-evolving transition matrices.
For finite-dimensional linear inverse problems where the variables are Gaussian, it is well-known that the minimum-mean-square error estimator takes the form of a regularized least-squares data fit. In this chapter, we show that this equivalence extends to a much broader infinite-dimensional setting where generalized splines take the role of linear regressors and generalized Gaussian processes on a nuclear space $S$ are the counterpart of Gaussian random vectors. The scope of this extension is of the same nature as the switch from the classic notion of function to that of a distribution, also known as a "generalized function." Our formalism involves a whitening/regularization operator $L: S\to S'$ whose continuous extension induces a native Hilbert space $H\subset S'$ that plays a central role in our characterization. The presentation is self-contained for the most part and remarkably general and powerful. It allows for the recovery of all known instances of such equivalences; in particular, the methods involving innovations and reproducing-kernel Hilbert spaces developed by Kailath and his students, and the mathematical correspondence between fractional splines and Mandelbrot's fractional Brownian motion (fractals), with the former being the optimal estimators of the latter. It also covers general Bayesian methods for the resolution of infinite-dimensional inverse problems.
Aleksandar Arandjelovic, Pavel V. Shevchenko, George Tzougasq-fin.RM stat.ML
Bayesian inference in compound loss models must often be repeated across policies, market scenarios, and prior specifications. Outside conjugate cases, this may require repeated numerical integration or Markov chain Monte Carlo (MCMC). We formulate this problem as amortized posterior approximation and construct a conditional Wasserstein generative adversarial network conditioned on sufficient statistics, prior mean and coefficient of variation, and mixture weights of prior families. Notably, a single shared generator is able to approximate the posterior laws of both the Poisson intensity and the Pareto shape parameter under mixtures of Gamma, inverse-Gaussian, and lognormal priors. We assess the approximation by simulation-based calibration and by comparisons with analytical posteriors, deterministic quadrature, and extensive MCMC simulations. In an application to data on extreme natural catastrophe losses, we produce rolling one-year posterior predictive distributions, and examine the effects of heavy-tailed severity and prior-family uncertainty on aggregate tail risk.
Many scientific and engineering applications require estimating unknown parameters from experimentally observable data -- an inverse problem that is inherently challenging due to nonlinearity, noise, and ill-posedness. In this paper, we propose an active diffusion-based inverse problem solver. A DM is trained to learn the mapping between the parameter space and the observable space. By iteratively detecting and correcting model misspecification through posterior uncertainty, the method discovers and learns the correct region of parameter space, even when initial training bounds exclude the true parameters. This provides a principled, Bayesian justification for adaptive domain augmentation and ensures robust inference for inverse problems under incomplete prior knowledge. We demonstrate the effectiveness of our inverse solver for a toy inverse problem with infinite solutions, and for the parameterization of the quantum correlation functions to event observables in a Quantum Chromodynamics analysis of nucleon structure.
Asir Intesar Tushar, Ioannis Sgouraliscs.LG stat.ML
Point-cloud data routinely captured by modern imaging and sensor technologies provide detailed geometric descriptions of objects and environments, but their analysis is hindered by large data volumes, localization noise, and missing information. In addition, existing point-cloud reconstruction pipelines typically return a single best-fit structure without uncertainty quantification. We introduce a fully Bayesian framework for representing point-cloud data and reconstructing closed curves, in which observed points are modeled as noisy perturbations of latent locations constrained to lie on the underlying curve that is regularized by a non-parametric prior. Posterior inference in our framework is carried out using a series of Markov chain Monte Carlo samplers tailored to point-cloud characteristics. Numerical experiments, including synthetic examples and real-world LiDAR datasets, show accurate reconstructions and quantified uncertainty over the recovered curves.
Exact Bayes prediction enjoys fast predictive regret guarantees, but exact posterior updating or representation may be too costly for online use. We study when these statistical guarantees are preserved by computational approximations. We show that the cumulative price of posterior approximation can be governed by the interaction between the contraction radius of the exact Gibbs posterior and the Wasserstein distance between the approximate and exact posteriors. Our general theorem shows that whenever exact Bayes prediction achieves a fast regret bound, any approximate posterior method that tracks the exact posterior with sufficient accuracy inherits the same fast regret, up to an additive term determined by the approximation error. Three online learning examples are developed. For linear models with strongly convex regularized losses, a projected Langevin algorithm yields an approximate posterior that achieves logarithmic regret. For an infinite-dimensional canonical exponential family sequence model over a Sobolev ellipsoid, a prior-preserving truncation method attains the minimax predictive regret rate with sublinear memory and constant update cost per observation. For random-design Gaussian process (GP) regression, a sparse variational posterior with inducing variables achieves the same predictive regret rate as the exact GP, but at substantially lower computational cost.
Many common data dependencies can be characterized by graphs: time series data are sequential (chain graph), images appear as pixels (lattice graph), areal data are defined by neighboring units (spatial adjacency graph), etc. Graph trend filtering seeks to smooth and predict such data. However, classical trend filtering only incorporates the graph for estimation of the trend, which limits its adaptivity, and is brittle in the presence of missing data. Further, it lacks uncertainty quantification and faces certain computing challenges. We address these limitations with a comprehensive Bayesian framework for (graph-) dependent data. Our approach leverages the graph at three critical junctures: 1) the trend, to enable smoothing, imputation, and prediction; 2) the local shrinkage, to enhance adaptivity and precision; and 3) the MCMC sampling algorithm, to deliver scalable posterior (predictive) inference via sparse and banded operations. For the proposed graph-dependent shrinkage priors, we study the local concentration and adaptivity properties and establish conditions for posterior propriety. Simulation studies demonstrate that, relative to state-of-the-art frequentist and Bayesian alternatives, this framework provides more accurate point estimates, more precise interval estimates, and highly competitive computing. We apply our methods for spatio-temporal modeling and forecasting of local area unemployment data for every county in the continental U.S. during the 2020 COVID-19 unemployment shock.
Yichen Zang, Song Liu, Jiun-Yi Linstat.ML cs.LG stat.ME
Simulation-Based Inference (SBI) serves as a vital framework for parameter inference in scientific fields where simulators involve intractable likelihoods, yet while amortized generative models offer rapid posterior estimation, they are often restricted by the specific priors used during training, thereby limiting their flexibility as prior knowledge evolves. To address this prior dependency, PriorGuide was introduced as an inference-time guidance method, but due to its intractable formulation, it relies on Gaussian approximations of the reverse transition kernel and Gaussian mixture model fitting for the prior ratio, both of which introduce systematic bias. Motivated by these limitations, we propose an unbiased test-time guidance framework that leverages Density Ratio Estimation (DRE) to learn a score guidance term, effectively decoupling the inference process from the prior training. Moreover, our framework remains agnostic to the specific density ratio estimators, making it a general and flexible framework for handling prior changes. Experimental results across multiple tasks demonstrate that our method matches or outperforms PriorGuide on C2ST and MMD in most tasks while maintaining robustness even under limited overlap between the training and target priors. Furthermore, we apply our method to Bayesian updating for parameter inference from planetary light-curve data, where it also demonstrates strong effectiveness and robustness. Code is available at https://github.com/a-chenchen/dre-based-prior-guidance .
Human motion forecasters are increasingly accurate and fast, but reliable deployment requires uncertainty estimates that are structured, calibrated, and efficient. Bayesian and ensemble-based uncertainty estimates often require repeated stochastic inference [15, 26], while conformal calibration alone does not provide an epistemic signal or preserve trajectory covariance structure [14, 50]. We introduce SPARC (Single-Pass Adaptive Risk Calibration), a Bayesian-conformal uncertainty layer for motion forecasting. A deterministic MLP backbone predicts the future mean, and a conjugate Bayesian last layer converts time-domain feature leverage into an analytic horizon-wise epistemic scale $κ_t(x)$. This scale inflates a graph-temporal Gaussian covariance without changing its correlation structure, and split conformal calibration produces 95% marginal prediction tubes with finite-sample validity under exchangeability. The key interface is the structured factorization $κ_t(x)Σ_{\mathrm{str},t}(x)$, which injects feature-space epistemic uncertainty into trajectory densities without Monte Carlo sampling. Across nine dataset-protocol blocks and deterministic, multimodal, and calibration baselines, SPARC ranks first on NLL and on the combined MPJPE+NLL criterion while retaining competitive point accuracy and efficient calibrated tubes. Ranking windows by $κ$ separates high-error cases, making the scale usable as a lightweight risk monitor.
Grégoire Sergeant-Perthuis, Elias Tsigaridas, Jules Tsukaharacs.LG cs.SC math.AG stat.ML
Classical information criteria such as the Bayesian Information Criterion (BIC) rely on regularity assumptions that break down for singular models, leading to incorrect model selection in settings such as deep learning. The Widely Applicable Bayesian Information Criterion (WBIC) relies on local learning coefficients $λ$, which in the analytic case coincides with local Real Log Canonical Thresholds (RLCT) of the Kullback-Leibler divergence of the model, to capture correct marginal likelihood asymptotics. Exact computation of the learning coefficients has been limited to special cases, and only sampling-based estimation methods are generally applicable. We present the first deterministic algorithm that computes local RLCTs exactly for any two-dimensional model whose Kullback-Leibler distance is contact equivalent to a polynomial, derive a bound on its complexity, and demonstrate its effectiveness for a broad class of models, with applications including polynomial neural networks. Beyond providing ground truth to calibrate sampling-based estimators, exact computation reveals algebraic structure in learning coefficients that sampling cannot and out-speeds it in the shallow regime.
Generative models have shown strong potential for positron emission tomography (PET) image reconstruction. Although diffusion model-based reconstruction methods have demonstrated promising performance, they often require many reverse sampling steps with data-consistency updates incorporated into the sampling process. Flow matching offers an attractive alternative because it can directly estimate clean images from intermediate states, allowing data-consistency refinement to be separated from flow propagation. In this work, we proposed flow matching-based PET image reconstruction methods. We first established PET-FlowDPS by incorporating Poisson likelihood guidance with an expectation-maximization (EM)-based preconditioner into the FlowDPS framework. We then proposed a model-based PET reconstruction method that used a pretrained flow matching model as a prior, in which the flow-based prior, PET data refinement, and stochastic propagation were interpreted within an approximate Bayesian framework. Experimental results using [$^{\text{18}}\text{F}$]FDG brain PET datasets showed that the proposed method achieved better bias-variance trade-offs across different dose levels compared with other reference methods. These results demonstrated the potential of flow matching as a generative prior for quantitative PET image reconstruction.
This paper presents a Bayesian control framework that integrates spike-based dynamics with probabilistic inference for adaptive control. Bayesian inference is widely regarded as a core computational principle of brain function, providing a normative framework for perception, decision-making, and learning under uncertainty. By combining a biologically inspired spiking neural model with Bayesian inference principles, we propose a brain-like control algorithm capable of operating in uncertain environments. We use the mountain car parking problem as a benchmark with non-linear dynamics. Our results demonstrate that the proposed controller can successfully update states in real time and generate goal-directed action plans through spike-driven dynamics. The results highlight the proposed model's potential as a bridge between computational neuroscience and probabilistic control theory.
Bayesian inference via Markov Chain Monte Carlo (MCMC) provides effective parameter estimation, but its real-time application in complex physical systems is hindered by heavy computational bottlenecks and extreme sensitivity to statistical noise. We address this by proposing a neural-network-based probabilistic surrogate framework for rapid and robust MCMC inference. Using fusion plasma Thomson scattering diagnostics as a challenging, noise-dominated testbed, our approach employs a dual-head architecture to simultaneously estimate the expected physical emission spectrum and the channel-wise intrinsic measurement noise variance. By optimizing a Gaussian Negative Log-Likelihood (GNLL) objective, the learned aleatoric uncertainty dynamically buffers the sampler against pathological shot noise. Evaluations demonstrate that this surrogate framework achieves > 1500x acceleration over exact physical forward models, while simultaneously reducing inference error (RMSE) by >20% compared to standard homoscedastic neural baselines, offering a highly promising paradigm for real-time physical analysis.
Despite the wide variety of existing Buy-`Til-You-Die (BTYD) models, nearly all rely upon the convenient assumption of transactions following a Poisson process. As modern customer bases grow larger and more diverse, a major gap in the marketing literature is BTYD models that can account for heterogeneity in timing patterns across millions of customers. This paper addresses that gap, introducing a family of models that assume transactions follow a Weibull renewal process and developing a highly scalable scheme for parameter estimation based on an amortized variational inference procedure. The proposed model fits to a proprietary dataset of 5 million online retail customers in 8 minutes which would take the current state-of-the-art an estimated 3-4 days. We show both theoretically and empirically that this dramatic improvement in computational performance comes with no appreciable change to either model interpretation or predictive performance. Beyond scalability, gradient-based variational inference also makes it easy to extend the model to covariates, which we illustrate on a public dataset of 4 million political donors during the 2020 US General election cycle. More generally, this paper demonstrates how to blend recent advances in approximate Bayesian inference and the tools of modern machine learning to dramatically improve the efficiency and expressivity of probabilistic models for customer base analysis.
This short work describes an extension of the permanental process model which includes fixed effects. By starting with a prior on the fixed effects coefficients we show that, in the diffuse prior limit, the intensity function of the permanental process can be found using the representer theorem and naturally decomposed into a fixed effects term and a function which is an element of a Reproducing Kernel Hilbert Space (RKHS). We show that the limiting equivalent kernel defines an RKHS whose squared norm is exactly the limiting penalty. This allows for straightforward scientific interpretation of permanental process models and the easy incorporation of domain knowledge into the estimation process.
How cautious should an agent be while it is still learning its environment? We propose RATTL (Risk-Adversarial Total-Reward Learning), which ties caution to epistemic uncertainty: the agent holds a Bayesian posterior over unknown dynamics and plans against a Wasserstein ambiguity set whose radius is a monotone function of that posterior. The radius contracts with evidence, so behaviour interpolates continuously between worst-case robustness and risk-neutral total-reward maximization. The design follows the duality underlying the Entropic Value-at-Risk, which converts the choice of a risk level into the choice of an ambiguity radius. We show the resulting planning problem is well posed under transience and compactness conditions, and prove a Safety Sandwich: the RATTL value lies between the uninformed robust value and the full- knowledge optimum, with a gap that vanishes as the posterior concentrates. In a canonical binary-hazard instance, the induced criterion reduces to Conditional Value-at-Risk at a level set by the posterior entropy. A worked example shows the agent deferring the efficient action until a sharp identification threshold. RATTL targets runtime safety for agents, including LLM-based systems, acting under uncertainty.
Ondrej Bajgar, Peter Tisnikar, Alessandro Abate +2cs.LG
The development of safe and beneficial AI requires that systems can learn and act in accordance with human preferences. However, explicitly specifying these preferences by hand is often infeasible. Inverse reinforcement learning (IRL) addresses this challenge by inferring preferences, represented as reward functions, from expert behaviour. We introduce Q-based Variational IRL (QVIRL), a novel Bayesian IRL method that recovers a posterior distribution over rewards from expert demonstrations via primarily learning a variational distribution over optimal Q-values. Unlike previous approaches, QVIRL combines scalability with uncertainty quantification, important for safety-critical applications as well as active learning. We demonstrate QVIRL's strong performance in apprenticeship learning across various tasks, including gridworlds, Lunar Lander, the Highway Environment, and two ATARI games both with static expert data and with active learning. It is the first method for Bayesian IRL that demonstrates training from raw pixel observations.
Machine learning (ML) is a key technology driving innovation today, but ensuring ML safety remains a major challenge for safety-related applications. A promising idea is to build proven-in-use arguments from field data, e.g. by running ML components (MLCs) in shadow mode or within safety envelopes so that their outputs can be monitored as 'safe probes' without affecting safety. These probes can then be used to build a statistical argument about field performance in a Bayesian way. However, many Bayesian field-data approaches in safety engineering model failures as a simple Bernoulli (or binomial) process with a single global failure probability and i.i.d. trials, which is rarely adequate for MLCs whose performance depends strongly on context. Statistical evidence is also about coverage of relevant situations, including edge cases, and building a single integrated statistical model for the entire system is usually not feasible. To address these challenges, this paper introduces CUBICS, a context-modular framework for per-component, situation-aware performance estimation of safety-relevant ML components. CUBICS partitions the operational design domain into situations and, for each safety-relevant component, defines a set of situation-specific assumptions and probabilistic guarantees that are represented and updated in a Bayesian manner using Subjective Logic (SL). By combining these guarantees with beliefs about how often each situation occurs, CUBICS derives an overall risk estimate for each component without requiring a monolithic system-level statistical model, and thus provides a building block for modular, field-data based safety assurance.
Harini Venkatesan, Christian Shelton, Ming-Feng Ho +2cs.AI
Markov chain Monte Carlo (MCMC) requires only the ability to evaluate the likelihood, making it a common technique for inference in complex models. However, it can have a slow mixing rate, requiring the generation of many samples to obtain good estimates and an overall high computational cost. FLARE MCMC is a multi-fidelity layered MCMC method that exploits lower-fidelity approximations of the true likelihood calculation to improve mixing and leads to overall faster performance. Such lower-fidelity likelihoods are commonly available in scientific and engineering applications where the model involves a simulation whose resolution or accuracy can be tuned. Our technique uses recursive, layered chains with simple layer tuning; it does not require the likelihood to take any form or have any particular internal mathematical structure. We demonstrate experimentally that FLARE MCMC achieves larger effective sample sizes for the same computational time across different scientific domains including hydrology and cosmology.
Cixuan Zhang, Guy Van den Broeck, Benjie Wangcs.LG cs.AI stat.ML
Causal discovery aims to uncover the underlying causal relationships given data generated from a system. The goal, however, is not merely to predict causal edges given data, but also to be able to interpret and explain either observed or hypothesized phenomena, such as a particularly large causal effect. We consider this task of conditional causal discovery and cast it as a Bayesian inference problem, in which we target the posterior over causal graphs and parameters conditional on an event such as a causal-effect constraint. Unfortunately, this poses a computational challenge: existing approaches to Bayesian causal discovery struggle when the event has small posterior mass. To address this, we adapt rare-event estimation techniques to perform inference the joint graph-parameter space. Our method gradually drives a particle population toward the constrained region while maintaining samples that approximate the conditional posterior. Empirical evaluation on synthetic graphs validates the accuracy of our approach at small and large scales, and we show in a case study on the Sachs protein dataset how our method can be used to aid scientific exploration by providing pathway-level summaries.
M. D. Champneys, M. R. Jones, A. J. Hughes +3eess.SP cs.LG
The robust treatment of environmental and operational variability (EOV) is an open challenge in population-based structural health monitoring (PBSHM). The difficulty is compounded in the case that the EOV signals are unmeasured. A common approach in conventional SHM is to apply \emph{projection-based} methods that discard subspaces of healthy feature data, reasoning that the EOV signal dominates the variance of the measured features. However, a common pitfall of projection-based approaches is that when damage acts close to the same variance-dominant direction, damage sensitivity is removed along with the EOV. An alternative identifying assumption for the removal of particular unmeasured EOVs is slowness; the latent EOV process is characterised by its long temporal correlation. In this paper, the latent EOV is cast as a state-space Gaussian process, enabling tractable $\mathcal{O}(T)$ inference via a Kalman filter. A robust hierarchical Bayesian identification framework is developed that enables population-level identification of latent EOVs and EOV-free residual features, using a Laplace approximation. The approach is first validated on a single laboratory-scale benchmark structure from the literature, subject to thermal EOVs, demonstrating robust damage detection and EOV recovery. The method is then applied to a simulated nine-turbine offshore wind farm with staggered deployment and damage, where it delivers a substantial true-positive uplift over projection and cointegration-based baselines at matched false-positive rates.
We study posterior contraction in positive-order Sobolev norms and Bayesian derivative estimation for infinite-dimensional exponential families. We embed the natural parameter in a Hilbert scale and model it via a standard Gaussian series prior expanded in the eigenbasis generating the scale. Under a two-sided link condition on the Fisher information and suitable local regularity assumptions, we show that smoothness-matching priors achieve minimax-optimal posterior contraction rates in any Hilbert scale norm up to the regularity of the ground truth. Our analysis builds on the novel approach to posterior contraction based on the Wasserstein distance recently introduced by Dolera et al. (2024). It combines refined Laplace-type estimates for infinite-dimensional integrals associated to the posterior kernels with a mixed-geometry estimate controlling their stability under fluctuations in the data, itself resting on a tailored Poincaré inequality for posterior distributions conditioned on neighbourhoods of the truth. We apply the general theory to density estimation with a logistic parametrisation, Poisson intensity estimation with an exponential link, and the Gaussian white-noise model, yielding minimax contraction rates in Sobolev norms across all three settings. In particular, these yield optimal recovery of density score functions and derivatives of Poisson intensities.
Predicting the answer to interventional ``what if'' questions --- the outcome of an action never taken --- requires a \emph{mechanistic}, causal model, not a curve fit; and learning such a model requires \emph{experiments}, because passive data leaves its mechanisms unidentified. Experiments are expensive, so the central problem is \emph{data efficiency}. We present the Model Discovery Agent (MDA), which couples a large language model (LLM), used as a \emph{proposer} of candidate structures, with standard Bayesian machinery --- sequential Monte Carlo (SMC) for parameter and structure posteriors, simulation-based inference (SBI) for intractable likelihoods, and value-of-information (VoI) for experiment design --- to discover latent mechanistic world models from few interventions. MDA operates in the M-open setting: when the truth lies outside the current hypothesis class, a predictive check flags the inadequacy and the proposer expands the hypothesis space with a new model whose parameters are then identified by designed experiments. We show that \emph{discovery and design reinforce}: the design step identifies the mechanism the discovery step proposes, and the identified mechanism improves predictions, enabling further discoveries from the remaining unexplained residuals. On three different benchmarks --- covering physics (\DPbench, \citep{wiemann2026discoverphysics}), chemistry (\CHEMbench, \citep{kabra2026autoscilab}) and biology (\HHbench, a new partially observed single-neuron electrophysiology benchmark we create) --- we show that MDA sets a new SOTA in terms of data-efficient model learning and reliable interventional forecasting ability.
Oussama Boussif, Mohammed Mahfoud, Younesse Kaddar +6cs.LG
Symbolic regression is the problem of finding an algebraic expression describing a stochastic dependence of a target variable on a set of inputs. Unlike forms of regression that fit parameters assuming a fixed model structure, symbolic regression is a search problem over the space of expressions, represented, for example, as abstract syntax trees using a library of operators. Symbolic regression is typically used in settings with limited, noisy data in the natural sciences. However, searching for a single best-fitting expression fails to capture the epistemic uncertainty about the expression, which motivates a Bayesian perspective that enables uncertainty quantification and specification of natural priors to constrain the search space. In this work, we propose ERRLESS (Entropy-Regularized Reinforcement Learning for Expression Structure Sampling), a scalable approach for sampling from the posterior distribution over expressions given data using maximum-entropy reinforcement learning. ERRLESS learns a neural policy that constructs expressions sequentially by building up their abstract syntax trees. At convergence, the policy samples expressions from the posterior. At test time, expressions can be sampled by rollouts of this policy. We demonstrate that ERRLESS achieves competitive results on the Feynman benchmark while producing short and interpretable expressions. Additionally, we demonstrate that the mean of the posterior predictive approximated by ERRLESS achieves a high coefficient of determination ($R^2$) compared to an SMC baseline, highlighting the benefits of the Bayesian perspective in symbolic regression.