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General Machine Learning · Causality

Hugh Dance, Peter Orbanz, Arthur Gretton

Reliable uncertainty quantification for causal effects is crucial in various applications, but remains difficult in nonparametric models, particularly for continuous treatments. We introduce IMPspec, a Gaussian process (GP) framework for modeling uncertainty over interventional causal functions under continuous treatments, which can be represented using reproducing Kernel Hilbert Spaces (RKHSs). By using principled function class expansions and a spectral representation of RKHS features, IMPspec yields tractable training and inference, a spectral algorithm to calibrate posterior credible intervals, and avoids the underfitting and variance collapse pathologies of earlier GP-on-RKHS methods. Across synthetic benchmarks and an application in healthcare, IMPspec delivers state-of-the-art performance in causal uncertainty quantification and downstream causal Bayesian optimization tasks.

Harry Zhang, Luca Carlone

Multimodal large language models (MLLMs) are playing an increasingly important role across multiple domains. In many applications, such as robotics, it is crucial to quantify the uncertainty in the output of these models. } We develop Fused Uncertainty with Semantic Evidence (FUSE), a probabilistic framework for capturing two complementary sources of uncertainty in multimodal language modeling: (i) aleatoric embedding-level uncertainty derived from input data vision-language ambiguity, and (ii) epistemic model-level uncertainty estimated from the semantic response diversity of MLLMs. Our approach formulates a Bayesian fusion mechanism that analytically combines these uncertainty sources to produce a scalar measure of uncertainty. This measure serves as a novel uncertainty representation for downstream applications of MLLMs and provides a principled foundation for uncertainty calibration in multimodal systems, improving reliability and downstream performance in MLLM-based reasoning and vision-language tasks. We demonstrate that our method outperforms baselines in providing uncertainty estimates and achieves state-of-the-art uncertainty calibration.

Probabilistic Methods · Monte Carlo and Sampling Methods

Youguang Chen, George Biros

Independent Metropolis–Hastings (IMH) algorithms are widely used in Bayesian inference, but their efficiency deteriorates when proposal distributions are constructed from inaccurate or approximate models. We introduce Proximal-IMH, an IMH method that enhances proposal distributions through a proximal posterior correction. Given an approximate posterior sample, each proposal is generated by minimizing a quadratically regularized surrogate objective, producing a local correction that balances fidelity to the exact model with stability around the approximate state. We analyze the resulting proposals from an optimization and probabilistic perspective, showing how the proximal correction improves alignment between approximate and exact posteriors and leads to improved acceptance and mixing behavior. The proposed framework applies to both linear and nonlinear forward operators, and is particularly well suited to Bayesian inverse problems where exact posterior sampling is computationally prohibitive. Numerical experiments on inverse problems with approximate forward models, including nonlinear operators, demonstrate that Proximal-IMH consistently outperforms existing IMH variants while retaining their simplicity and scalability.

Probabilistic Methods · Gaussian Processes

Wenyuan Zhao, Rui Tuo, Chao Tian

Gaussian processes (GPs) provide a principled Bayesian framework for uncertainty estimation, but their computational complexity severely limits scalability to large datasets. We propose SIKA-GP, which accelerates GP inference using sparse inducing kernel approximations based on a dyadic ordered template basis, incurring only ${O}(\log M)$ complexity dependence on the number of inducing points. Our approach constructs compact and expressive kernel representations from sparsely activated bases, enabling efficient tensorized GPU computation and seamless integration with modern large-scale models. SIKA-GP can be naturally embedded into Bayesian neural networks (BNNs) with sparse activations, yielding significant speedups in both training and inference without sacrificing predictive performance. The method naturally extends to deep feature learning, addressing the scalability challenges introduced by deep architectures and high-dimensional feature representations. Empirical results on vision and transformer-based language benchmarks demonstrate that our approach consistently delivers fast and accurate GP models, providing a principled path toward scalable kernel learning.

Probabilistic Methods · Everything Else

Sammy Sharief, Justine Zeghal, Gabriel Missael Barco, Pablo Lemos, Yashar Hezaveh, Laurence Perreault-Levasseur

We present Mira, a method for estimating the expected probability that samples from a candidate conditional distribution match the true, unknown conditional distribution, for which only data-label pairs are available. We derive theoretical bounds obtained when the candidate distribution matches the true one and when the conditional distributions are independent. This framework thus enables model comparison by quantifying the alignment between the conditional distribution of a candidate model and the data-label pairs of the true model. Consequently, Mira enables Bayesian model comparison through direct posterior validation, bypassing the challenging evidence computation. We demonstrate its effectiveness across several toy problems and Bayesian inference tasks.

Probabilistic Methods · Bayesian Models and Methods

Arjun Mani, Carl Vondrick, Richard Zemel

Many real-world design problems involve optimizing an expensive black-box function $f(x)$, for which Bayesian Optimization is a sample-efficient framework. However, while the basic black-box setting returns a scalar reward, real-world experiments often generate a wealth of useful information. We introduce a new setting where an experiment generates high-dimensional auxiliary information $h(x)$ along with $f(x)$; moreover, a history of relevant, previously-solved tasks is available for accelerating optimization. We develop a novel method based on a neural model which predicts $f(x)$ for unseen designs given a few-shot context containing observations of $h(x)$. We evaluate our method on two challenging domains, robotic hardware design and hyperparameter tuning. On both domains, our method achieves improved few-shot prediction and faster design optimization, outperforming several multi-task optimization methods.

Probabilistic Methods · Bayesian Models and Methods

Zerui Tao, Qibin Zhao

Low-rank tensor decomposition (TD) is usually effective on clean, fully observed data, but it often degrades under severe missingness or noise. The low-rank constraint alone provides a weak inductive bias, while common handcrafted priors (e.g., sparsity or smoothness) fail to capture rich real-world structures. 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 Bayesian CP decomposition framework that combines a cumulative shrinkage process prior for automatic rank selection with an off-the-shelf pre-trained diffusion model as an implicit 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.

Deep Learning · Large Language Models

Xuanze Zhao, Hongcheng Ding, Jing Jin, LIU XUANHUANG, Shamsul Abdullah, Deshinta Dewi

Full fine-tuning of large language models (LLMs) incurs prohibitive computational and storage costs. Parameter-efficient fine-tuning (PEFT) addresses this limitation, with Low-Rank Adaptation (LoRA) gaining widespread adoption due to its simplicity and zero inference overhead. However, LoRA and its variants typically rely on uniform rank allocation or a single importance metric such as gradient magnitude or output sensitivity to guide rank distribution. This approach fails to recognize that gradient magnitude and output contribution are decoupled properties, leading to suboptimal allocation where critical layers are under-provisioned while less important ones waste capacity. To address this challenge, we propose COBRA, a principled framework integrating dual importance factors for adaptive rank allocation. COBRA operates in three stages: (1) layer conductance attribution quantifies each layer's contribution via path-integral attribution; (2) dual-factor aggregation combines contribution with adaptation demand, producing the TA-LC distribution; and (3) Bayesian rank allocation translates this distribution into optimal heterogeneous ranks via variational optimization. Layer conductance provides layer-level interpretability by explicitly quantifying how much each layer contributes to predictions without redundancy, directly aligning with the granularity of rank allocation decisions and enabling principled cross-layer comparison for rank distribution. Experiments across diverse architectures and tasks demonstrate that COBRA consistently outperforms existing methods, achieving up to 1.6 points improvement on GLUE and 6.6\% average gain in high-rank regimes under comparable parameter budgets.

Probabilistic Methods · Bayesian Models and Methods

Tom Sprunck, Marcelo Pereyra, Tobías Liaudat

Modern imaging techniques heavily rely on Bayesian statistical models to address difficult image reconstruction and restoration tasks. This paper addresses the objective evaluation of such models in settings where ground truth is unavailable, with a focus on model selection and misspecification diagnosis. Existing unsupervised model evaluation methods are often unsuitable for computational imaging due to their high computational cost and incompatibility with modern image priors defined implicitly via machine learning models. We herein propose a general methodology for unsupervised model selection and misspecification detection in Bayesian imaging sciences, based on a novel combination of Bayesian cross-validation and data fission, a randomized measurement splitting technique. The approach is compatible with any Bayesian imaging sampler, including diffusion and plug-and-play samplers. We demonstrate the methodology through experiments involving various scoring rules and types of model misspecification, where we achieve excellent selection and detection accuracy with a low computational cost.

Applications · Computer Vision

Xing Xi, Yu Qiu, Ronghua Luo, Peixian Chen, peilin tong, Jiahao Liang

Recent progress in vision-language models (VLMs) has driven significant advances in video understanding. However, existing methods often act as naive empiricists, mapping video input directly to output without any mechanism to introspect or challenge inherent bias. In this work, we challenge this paradigm by reframing video reasoning as a Bayesian cognitive process. We propose Video-BCI (Bayesian Cognitive Integration of Self-Prior Hypotheses), a novel framework that first samples a set of Self-Prior Hypotheses to represent the model's intuitive yet potentially biased cognitive state, and then guides the VLMs to perform a critical integration of these priors. This process encourages the model to challenge erroneous majority consensus in cases of high information divergence and to distill superior reasoning chains from its own prior space. The integration is driven by a composite Cognitive Utility Function comprising two intrinsic learning signals: Dialectical Uncertainty Signal (DUS) and Process Tracing Signal (PTS). The DUS incentivizes correct, non-majority judgments by quantifying both the conflict (entropy) among priors and their consensus-challenging strength. The PTS guides the model to trace and learn from reasoning paths within its own priors that lead to better answers, enabling self-driven procedural knowledge distillation. Extensive experiments on six mainstream benchmarks show that Video-BCI achieves new state-of-the-art (SOTA) results across the board. For example, it surpasses the previous best on the MMVU benchmark by 3.8%. Our code will be made publicly available.

Probabilistic Methods · Bayesian Models and Methods

Zijian Chen, Archana Venkataraman

Human ratings are central to learning and inference across several application domains, but they are also subject to inter-rater biases and judgment errors. Quantifying the uncertainty of these human ratings would require repeated measurements, which are expensive and rarely available at scale. We propose a Bayesian graphical model to estimate the instance-level and item-level uncertainty of (subjective) human ratings by leveraging auxiliary (objective) data. Our model learns a shared latent content representation that explains factors common to both the human rating and auxiliary data and a latent uncertainty variable that captures fluctuations in the human assessments via a data-conditioned prior. We develop a scalable amortized variational inference procedure that uses modality-appropriate neural encoders and decoders to represent the posterior factors. Experiments on synthetic data demonstrate that our framework can accurately recover the latent uncertainty under targeted ablations and stress tests. We further demonstrate our approach on a real-world dataset of paired functional MRI scans and behavioral testing for autism, thus highlighting the need for uncertainty quantification.

Optimization · Discrete and Combinatorial Optimization

Esha Singh, Dongxia Wu, Chien-Yi Yang, Tajana Rosing, Rose Yu, Yian Ma

Multi-objective combinatorial optimization seeks Pareto-optimal solutions over exponentially large discrete spaces, yet existing methods sacrifice generality, scalability, or theoretical guarantees. We reformulate it as an online learning problem over a decomposed decision space, solving position-wise bandit subproblems via adaptive expert-guided sequential construction. This formulation admits regret bounds of $O(d\sqrt{T \log T})$ depending on subproblem dimensionality \(d\) rather than combinatorial space size. On standard benchmarks, our method achieves 80--98\% of specialized solvers performance while achieving two to three orders of magnitude improvement in sample and computational efficiency over Bayesian optimization methods. On real-world hardware-software co-design for AI accelerators with expensive simulations, we outperform competing methods under fixed evaluation budgets. The advantage grows with problem scale and objective count, establishing bandit optimization over decomposed decision spaces as a principled alternative to surrogate modeling or offline training for multi-objective optimization.

Deep Learning · Generative Models and Autoencoders

Yida Xiong, Jiameng Chen, Xiuwen Gong, Jia Wu, Shirui Pan, wenbin Hu

Graph generation aims to sample discrete node and edge attributes while satisfying coupled structural constraints. Diffusion models for graphs often adopt largely factorized forward-noising, and many flow-matching methods start from factorized reference noise and coordinate-wise interpolation, so node–edge coupling is not encoded by the generative geometry and must be recovered implicitly by the core network, which can be brittle after discrete decoding. Bayesian Flow Networks (BFNs) evolve distribution parameters and naturally support discrete generation. But classical BFNs typically rely on factorized beliefs and independent channels, which limit geometric evidence fusion. We propose Variational Bayesian Flow Network (VBFN), which performs a variational lifting to a tractable joint Gaussian variational belief family governed by structured precisions. Each Bayesian update reduces to solving a symmetric positive definite linear system, enabling coupled node and edge updates within a single fusion step. We construct sample-agnostic sparse precisions from a representation-induced dependency graph, thereby avoiding label leakage while enforcing node–edge consistency. On synthetic and molecular graph datasets, VBFN improves fidelity and diversity, and surpasses baseline methods. The reproducible code is available at https://anonymous.4open.science/r/VBFN-D6B0/.

Deep Learning · Large Language Models

Tim Tomov, Dominik Fuchsgruber, Stephan Günnemann

In many applications of LLMs, natural language responses often have an underlying structure such as representing discrete labels, numerical values, or graphs. Yet, existing decoding and uncertainty estimation methods operate only in language space and largely disregard structural information. We address this by modeling LLM outputs directly in a task-dependent latent structure. By equipping this structure with a dissimilarity measure, we can compute Bayes-optimal responses. These are not selected from sampled generations but are newly synthesized by combining individual responses in the latent space. Across different tasks, Bayes-optimal responses consistently outperform standard decoding methods like beam search. Moreover, quantifying uncertainty via the induced Bayesian risk captures variations in terms of the latent structure and improves alignment with output quality and correctness. Our decision-theoretic framework is applicable to any problem that admits a latent response structure and enables reliable task-aware LLM predictions.

Probabilistic Methods · Bayesian Models and Methods

Niels Bracher, Lars Kühmichel, Desi Ivanova, Xavier Intes, Paul Buerkner, Stefan Radev

We consider problems of parameter estimation where design variables can be actively optimized to maximize information gain. To this end, we introduce JADAI, a framework that jointly amortizes Bayesian adaptive design and inference by training a policy, a history network, and an inference network end-to-end. The networks minimize a generic loss that aggregates incremental reductions in posterior error along experimental sequences without density evaluations. Inference networks are instantiated with diffusion models that can approximate high-dimensional and multimodal posteriors at every experimental step. JADAI achieves superior or competitive performance across adaptive design benchmarks.

Reinforcement Learning · Batch/Offline

Tianwei Ni, Esther Derman, Vineet Jain, Vincent Taboga, Siamak Ravanbakhsh, Pierre-Luc Bacon

Popular offline reinforcement learning (RL) methods rely on conservatism, penalizing out-of-dataset actions or restricting rollout horizons. We question the universality of this principle and revisit a complementary Bayesian perspective. By modeling a posterior over plausible world models and training a history-dependent agent to maximize expected return, the Bayesian approach directly addresses epistemic uncertainty and enables test-time generalization, without conservatism. We first illustrate in a bandit setting that Bayesianism excels on low-quality datasets where conservatism fails. Scaling to realistic tasks, we find that long-horizon rollouts are essential to control value overestimation once conservatism is removed. We introduce design choices that enable learning from long-horizon rollouts while mitigating compounding errors, yielding our algorithm, NEUBAY, grounded in the neutral Bayesian principle. On D4RL and NeoRL benchmarks, NEUBAY is competitive with leading conservative algorithms, achieving new state-of-the-art on 7 datasets with rollout horizons of several hundred steps. Finally, we characterize datasets by quality and coverage to identify when NEUBAY is preferable to conservative methods.

Probabilistic Methods · Graphical Models

Juha Harviainen, Frank Sommer, Manuel Sorge

Polytrees are a subclass of Bayesian networks that seek to capture the conditional dependencies between a set of $n$ variables as a directed forest and are motivated by their more efficient inference and improved interpretability. Since the problem of learning the best polytree is NP-hard, we study which restrictions make it more tractable by considering for example in-degree bounds, properties of score functions measuring the quality of a polytree, and approximation algorithms. We devise an algorithm that finds the optimal polytree in time $\mathcal{O}\big((2+\epsilon)^n\big)$ for arbitrarily small $\epsilon > 0$ and any constant in-degree bound $k$, improving over the fastest previously known algorithm of time complexity $\mathcal{O}\big(3^n\big)$. We further give polynomial-time algorithms for finding a polytree whose score is within a factor of $k$ from the optimal one for arbitrary scores and a factor of $2$ for additive ones. Many of the results are complemented by (nearly) tight lower bounds for either the time complexity or the approximation factors.

Probabilistic Methods · Bayesian Models and Methods

Yaohong Yang, Sammie Katt, Samuel Kaski

Multi-objective Bayesian optimization (MOBO) provides a principled framework for optimizing expensive black-box functions with multiple objectives. However, existing MOBO methods often struggle with coverage, scalability with respect to the number of objectives, and integrating constraints and preferences. In this work we propose STAGE-BO, Sequential Targeting Adaptive Gap-Filling $\varepsilon$-Constraint Bayesian Optimization, that explicitly targets under-explored regions of the Pareto front. By analyzing the coverage of the approximate Pareto front, our method identifies the largest geometric gaps. These gaps are then used as constraints, which transforms the problem into a sequence of inequality-constrained subproblems, efficiently solved via constrained expected improvement acquisition. Our approach provides a uniform Pareto coverage without hypervolume computation and naturally applies to constrained, and preference-based settings. Experiments on synthetic and real-world benchmarks demonstrate superior coverage and competitive hypervolume performance against state-of-the-art baselines.

Probabilistic Methods · Bayesian Models and Methods

Louis Mandel, Guillaume Baudart, Mandana Vaziri, Martin Hirzel

Building reliable applications that leverage large language models (LLMs) remains a significant challenge. While LLMs offer impressive capabilities across diverse tasks, their outputs often lack accuracy and provide no clear measure of confidence. This uncertainty compounds in flows of multiple calls to LLMs and other tools, making it difficult for developers and end-users to trust the results. This paper introduces a probabilistic language for programming LLM-based flows. It enables developers to quantify and propagate uncertainty throughout the application's flow, and experiment with different inference scaling techniques without adding a single line of code beyond the flow's logic. We present an experimental study to demonstrate this capability, and a case study building a theorem proving agent for the Rocq theorem prover.

Probabilistic Methods · Bayesian Models and Methods

Theodore Papamarkou, Pierre Alquier, Matthias Bauer, Wray Buntine, Andrew Davison, Gintare Karolina Dziugaite, Maurizio Filippone, Andrew Y. K. Foong, Vincent Fortuin, Dimitris Fouskakis 等

LLMs excel at predictive tasks and complex reasoning tasks, but many high-value deployments rely on decisions under uncertainty, for example, which tool to call, which expert to consult, or how many resources to invest. While the usefulness and feasibility of Bayesian approaches remain unclear for LLM inference, this position paper argues that the control layer of an agentic AI system (that orchestrates LLMs and tools) is a clear case where Bayesian principles should shine. Bayesian decision theory provides a framework for agentic systems that can help to maintain beliefs over task-relevant latent quantities, to update these beliefs from observed agentic and human-AI interactions, and to choose actions. Making LLMs themselves explicitly Bayesian belief-updating engines remains computationally intensive and conceptually nontrivial as a general modeling target. In contrast, this paper argues that coherent decision-making requires Bayesian principles at the level of the agentic system, not necessarily the LLM agent parameters. This paper articulates practical properties for Bayesian control that fit modern agentic AI systems and human-AI collaboration, and provides concrete examples and design patterns to illustrate how calibrated beliefs and utility-aware policies can improve agentic AI orchestration.