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Probabilistic Methods · Bayesian Models and Methods

Ibne Farabi Shihab, SANJEDA AKTER, Anuj Sharma

Deep protein structure predictors such as AlphaFold provide confidence estimates (e.g., pLDDT) that are not calibrated and degrade under distribution shifts across experimental modalities, temporal changes, and disordered regions. We introduce **CalPro**, a prior-aware evidential conformal framework for shift-robust uncertainty quantification. CalPro combines three components: (i) a geometric evidential head outputting Normal Inverse Gamma distributions via graph neural networks; (ii) a differentiable calibration surrogate that shapes representations during training, followed by split-conformal calibration for finite-sample coverage; and (iii) domain priors (disorder, flexibility) encoded as soft constraints on predicted uncertainty. Theoretically, we derive structure-aware *sensitivity bounds* for coverage degradation under distribution shift using PAC-Bayesian control over ambiguity sets, quantifying how miscoverage increases with model complexity and shift magnitude. Empirically, CalPro achieves at most 5 percentage points coverage degradation across modalities compared to 15 to 25 points for baselines, reduces calibration error by 30\% to 50\%, and improves downstream docking success from 52\% to 75\% when filtering by uncertainty. The framework extends beyond proteins to structured regression tasks where priors encode local reliability.

Probabilistic Methods · Variational Inference

Nico Daheim, Thomas Moellenhoff, James Ming Liang Ang, Mohammad Emtiyaz Khan

Stochastic Variance Reduced Gradient (SVRG) and its variants aim to speed-up training by using gradient corrections. In their decade of existence, these methods have never been connected to any Bayesian methods, at least not at a fundamental level. Here, we fill this gap and show surprising new connections of SVRG to a recently proposed Bayesian method called ‘posterior correction’. Our main contribution is to show that SVRG can be recovered as a special case of posterior correction when applied over isotropic-Gaussian posteriors. Novel extensions of SVRG are automatically obtained by using more flexible exponential-family posteriors. We derive two new such extensions by using Gaussian families: a Newton-like variant with novel Hessian corrections, and an Adam-like extension that scales to large problems. Our work is the first to connect SVRG to Bayes and use it to boost training.

Probabilistic Methods · Monte Carlo and Sampling Methods

Panagiotis Theodoropoulos, Juno Nam, Evangelos Theodorou, Jaemoo Choi

Transportation on graphs is a fundamental challenge across many domains, where decisions must respect topological and operational constraints. Despite the need for actionable policies, existing graph-transport methods lack this expressivity. They rely on restrictive assumptions, fail to generalize across sparse topologies, and scale poorly with graph size and time horizon. To address these issues, we introduce Generalized Schrödinger Bridge on Graphs (GSBoG), a novel scalable data-driven framework for learning executable controlled continuous-time Markov chain (CTMC) policies on arbitrary graphs under state cost augmented dynamics. Notably, GSBoG learns trajectory-level policies, avoiding dense global solvers and thereby enhancing scalability. This is achieved via a likelihood optimization approach, satisfying the endpoint marginals, while simultaneously optimizing intermediate behavior under state-dependent running costs. Extensive experimentation on challenging real-world graph topologies shows that GSBoG reliably learns accurate, topology-respecting policies while optimizing application-specific intermediate state costs, highlighting its broad applicability and paving new avenues for cost-aware dynamical transport on general graphs.

Probabilistic Methods · Everything Else

Kianoosh Ashouritaklimi, Stefano Cortinovis, Francois Caron

Bayes--assisted conformal prediction combines the strengths of Bayesian modelling with exact, distribution--free frequentist coverage guarantees. While validity holds even under model misspecification, the size of the prediction sets can degrade significantly when the prior is poorly aligned with the observed data. We address this limitation by introducing \textbf{RoBAS}: a novel Bayes--assisted nonconformity score which is motivated by a hierarchical Bayesian working model with heavy--tailed priors, and which we implement in practice via a computationally tractable empirical Bayes instantiation. Our proposed method is adaptive to the quality of the available working information in the prior. When reliable prior information is available and can be effectively encoded, we achieve set sizes lower than that of other sets with the same coverage. On the other hand, when such information is weak or inaccurate, our nonconformity scores revert to the Distance--To--Average score, a robust baseline that is well--suited to settings where accurate prior information is not available. We evaluate our method on tabular and image regression tasks in the setting where there exists distribution shift between the training and calibration/test data. We find that our approach is competitive with widely used nonconformity scores in the absence of distribution shift, while providing significant gains in the more challenging setting of distribution shift.

Probabilistic Methods · Everything Else

Yating Liu, Yeo Jin Jung, Zixuan Wu, So Won Jeong, Claire Donnat

Conformal prediction provides distribution-free prediction sets with finite-sample conditional guarantees. RKHS-based frameworks—while promising for complex covariate shifts—suffer from prohibitive computational costs. To guarantee conditional validity under such shifts while ensuring feasibility, we build upon the framework of (Gibbs et al., 2023) by introducing a stable and efficient algorithm that computes the full solution path of the regularized RKHS conformal optimization problem, at essentially the same cost as a single kernel quantile fit. Our approach provides simultaneous hyperparameter tuning which provides smoothness control and data-adaptive calibration. To extend the method to high-dimensional settings, we further integrate our approach with low-rank latent embeddings that capture conditional validity in a data-driven latent space. Empirically, our method provides reliable conditional coverage across a variety of modern black-box predictors, improving the interval length of (Gibbs et al., 2023) by 30%, while achieving a 40-fold speedup.

Probabilistic Methods · Variational Inference

Kyurae Kim, Qiang Fu, Yian Ma, Jacob Gardner, Trevor Campbell

For approximating a target distribution given only its unnormalized log-density, stochastic gradient-based variational inference (VI) algorithms are a popular approach. For example, Wasserstein VI (WVI) and black-box VI (BBVI) perform gradient descent in measure space (Bures-Wasserstein space) and parameter space, respectively. Previously, for the Gaussian variational family, convergence guarantees for WVI have shown superiority over existing results for black-box VI with the reparametrization gradient, suggesting the measure space approach might provide some unique benefits. In this work, however, we close this gap by obtaining identical state-of-the-art iteration complexity guarantees for both. In particular, we identify that WVI's superiority stems from the specific gradient estimator it uses, which BBVI can also leverage with minor modifications. The estimator in question is usually associated with Price's theorem and utilizes second-order information (Hessians) of the target log-density. We will refer to this as Price's gradient. On the flip side, WVI can be made more widely applicable by using the reparametrization gradient, which requires only gradients of the log-density. We empirically demonstrate that the use of Price's gradient is the major source of performance improvement.

Probabilistic Methods · Variational Inference

Bohan Wu, David Blei

Variational inference (VI) has emerged as a popular method for approximate inference for high-dimensional Bayesian models. In this paper, we propose a novel VI method that extends the naive mean field via entropic regularization, referred to as $\Xi$-variational inference ($\Xi$-VI). $\Xi$-VI has a close connection to the entropic optimal transport problem and benefits from the computationally efficient Sinkhorn algorithm. We show that $\Xi$-variational posteriors effectively recover the true posterior dependency, where the likelihood function is downweighted by a regularization parameter. We analyze the role of dimensionality of the parameter space on the accuracy of $\Xi$-variational approximation and the computational complexity of computing the approximate distribution, providing a rough characterization of the statistical-computational trade-off in $\Xi$-VI, where higher statistical accuracy requires greater computational effort. We also investigate the frequentist properties of $\Xi$-VI and establish results on consistency, asymptotic normality, high-dimensional asymptotics, and algorithmic stability. We provide sufficient criteria for our algorithm to achieve polynomial-time convergence. Finally, we show the inferential benefits of using $\Xi$-VI over mean-field VI and other competing methods, such as normalizing flow, on simulated and real datasets.

Probabilistic Methods · Everything Else

Herilalaina Rakotoarison, Steven Adriaensen, Tom Viering, Samuel Gabriel Müller, Carl Hvarfner, Frank Hutter, Eytan Bakshy

Information-theoretic acquisition functions such as Entropy Search (ES) offer a principled exploration–exploitation framework for Bayesian optimization (BO). However, their practical implementation relies on complicated and slow approximations, i.e., a Monte Carlo estimation of the information gain. This complexity can introduce numerical errors and requires specialized, hand-crafted implementations. We propose a two-stage amortization strategy that learns to approximate entropy search-based acquisition functions using Prior-data Fitted Networks (PFNs) in a single forward pass. A first PFN is trained to be conditioned on information about the optima; second, the α-PFN is trained to predict the expected information gain by training on information gains measured with the first PFN. The α-PFN offers a flexible learned approximation, which replaces the complex heuristic approximations with a single forward pass per candidate, enabling rapid and extensible acquisition evaluation. Empirically, our approach is competitive with state-of-the-art entropy search implementations on synthetic and real-world benchmarks, while accelerating the different entropy search variants across all our experiments, with speed ups over 50x.

Probabilistic Methods · Bayesian Models and Methods

Moule Lin, Shuhao Guan, Andrea Patane, David Gregg, Goetz Botterweck

Large Language Models usually put more emphasis on accuracy and therefore, will guess even when not certain about the prediction, which is especially severe when fine-tuned on small datasets due to the inherent tendency toward miscalibration. In this work, we introduce Bayesian-LoRA, which reformulates the deterministic LoRA update as a probabilistic low-rank representation inspired by Sparse Gaussian Processes. We identify a structural isomorphism (in the functional sense of shared bilinear form, not strict algebraic equivalence) between LoRA's factorization and Kronecker-factored SGP posteriors, and show that LoRA emerges as a limiting case when posterior uncertainty collapses. We conduct extensive experiments on various LLM architectures across commonsense reasoning benchmarks. With only approximately 0.42M additional parameters and ${\approx}1.2{\times}$ training cost relative to standard LoRA, Bayesian-LoRA significantly improves calibration across models up to 30B, achieving up to 84\% ECE reduction and 76\% NLL reduction while maintaining competitive accuracy for both in-distribution and out-of-distribution (OoD) evaluations.

Deep Learning · Algorithms

Mingsheng Cao, Hongliang Chen, Ming Hu, Fei Gao, Qiaolong Ding, Wenke Huang, Xiaofei Xie, Junlong Zhou

Although Federated Learning (FL) offers advantages in privacy-preserving for cross-device collaborative learning, its practical deployment remains severely constrained by heterogeneous hardware resources and non-IID (non-independent and identically distributed) data across devices. Sub-model extraction has emerged as a widely adopted strategy for enabling collaborative training among devices with heterogeneous models. However, existing sub-model extraction methods in FL typically rely on coarse-grained stochastic selection or rigid rule-based neuron selection, which severely limits training performance. Specifically, stochastic strategies lead to severe parameter conflicts under non-IID data distributions, while rule-based approaches lack diversity in neuron selection per device, preventing comprehensive parameter optimization. To address this problem, this paper presents a novel sub-model extraction-based FL framework, named SpineFL, which adopts a backbone-sharing mechanism and an activation-guided pruning strategy for sub-model extraction. Specifically, SpineFL decomposes each global model layer into two portions: i) a mandatory backbone shared by all the sub-models to maintain model generalization, and ii) a dynamic portion for sub-model extraction. SpineFL adopts the activation-guided selection strategy to probabilistically select neurons according to their activation frequency from the dynamic portion to generate sub-model, where neurons exhibiting higher historical activation are more likely to be included, thereby simultaneously addressing parameter conflicts while preserving selection diversity. Experimental results demonstrate that compared with state-of-the-art heterogeneous FL methods, SpineFL can achieve up to 3.28% accuracy improvement.

Probabilistic Methods · Bayesian Models and Methods

Armin Beck, Peter Ochs

Symmetries are known to improve the empirical performance of machine learning models, yet theoretical guarantees explaining these gains remain limited. Prior work has focused mainly on compact group symmetries and often assumes that the data distribution itself is invariant, an assumption rarely satisfied in real-world applications. In this work, we extend generalization guarantees to the broader setting of non-compact symmetries, such as translations and to non-invariant data distributions. Building on the PAC-Bayes framework, we adapt and tighten existing bounds, demonstrating the approach on McAllester's PAC-Bayes bound while showing that it applies to a wide range of PAC-Bayes bounds. We validate our theory with experiments on a rotated MNIST dataset with a non-uniform rotation group, where the derived guarantees not only hold but also improve upon prior results. These findings provide theoretical evidence that, for symmetric data, symmetric models are preferable beyond the narrow setting of compact groups and invariant distributions, opening the way to a more general understanding of symmetries in machine learning.

Probabilistic Methods · Bayesian Models and Methods

Nicolas M Zilberstein, Florentin Guth, Santiago Segarra, Eero Simoncelli

Generative diffusion models can provide powerful priors for inverse problems in imaging, but existing implementations suffer from two key limitations: $(i)$ they learn only an implicit approximation of the prior density, and $(ii)$ they rely on crude likelihood approximations that introduce biases in the sampling. We address these challenges by introducing a new energy-based model trained using denoising score matching with a covariance-based regularization that enforces consistency across different inverse problems. Our approach learns explicit, normalized posterior densities for diverse linear inverse problems using a single model, while preserving the sampling capabilities of diffusion models. This enables new capabilities unavailable to score-based diffusion models: energy-guided adaptive sampling that adjusts schedules on-the-fly, unbiased MCMC correction with Metropolis-Hastings acceptance, and blind degradation estimation via Bayes rule. We validate our method on multiple datasets (MNIST, CelebA, ImageNet) and tasks (inpainting, deblurring), demonstrating competitive or superior performance to established baselines.

Probabilistic Methods · Everything Else

Tanish Agarwal, Kaustubh Shivshankar Shejole, Arpit Agarwal

Item response theory (IRT) is widely used in areas such as recommender systems, education, psychology, and other fields. A popular model for IRT is the Rasch model. Under this model, if a user with ability $\theta$ performs a task with difficulty $\beta$ then its label $X \sim \text{Bernoulli} (1 / (1 + \exp(-(\theta - \beta)))$. Existing joint maximum likelihood estimation approaches for this problem do not perform well on small datasets and also lack theoretical guarantees. Recently, Nguyen and Zhang proposed a two step approach: (1) spectral method for estimation of task parameters, (2) likelihood optimization for estimation of user parameters. While this approach is theoretically sound, it is not computationally efficient. In this work, we propose an EM-based algorithm for joint estimation of item and user parameters by introducing Pólya-Gamma latent variables, which simplify the logistic log-likelihood. We show that our algorithm is both theoretically sound and consistently outperforms existing methods on synthetic and real-world datasets.

Probabilistic Methods · Bayesian Models and Methods

Sara Pérez-Vieites, Sahel Iqbal, Simo Särkkä, Dominik Baumann

Bayesian experimental design (BED) provides a principled framework for optimizing data collection by choosing experiments that are maximally informative about unknown parameters. However, existing methods cannot deal with the joint challenge of (a) *partially observable dynamical systems*, where only noisy and incomplete observations are available, and (b) *fully online inference*, which updates posterior distributions and selects designs sequentially in a computationally efficient manner. Under partial observability, dynamical systems are naturally modeled as state-space models (SSMs), where latent states mediate the link between parameters and data, making the likelihood---and thus information-theoretic objectives like the expected information gain (EIG)---intractable. We address these challenges by deriving new estimators of the EIG and its gradient that explicitly marginalize latent states, enabling scalable stochastic optimization in nonlinear SSMs. Our approach leverages nested particle filters for efficient online state-parameter inference with convergence guarantees. Applications to realistic models, such as the susceptible–infectious–recovered (SIR) and a moving source location task, show that our framework successfully handles both partial observability and online inference.

Probabilistic Methods · Bayesian Models and Methods

Jiaxiang Yi, Miguel Bessa

Real-world data contains aleatoric uncertainty -- irreducible noise arising from imperfect measurements or from incomplete knowledge about the data generation process. Mean variance estimation (MVE) networks can learn this type of uncertainty but require ad-hoc regularization strategies to avoid overfitting and are unable to predict epistemic uncertainty (model uncertainty). Conversely, Bayesian neural networks predict epistemic uncertainty but are notoriously difficult to train due to the approximate nature of Bayesian inference. We propose to cooperatively train a variance estimation network with a Bayesian neural network and empirically demonstrate that the resulting model disentangles aleatoric and epistemic uncertainties while improving the mean estimation. We demonstrate the effectiveness and scalability of this method across a diverse range of datasets, including a time-dependent heteroscedastic regression dataset we created where the aleatoric uncertainty is known, used to assess estimation accuracy. The proposed method is straightforward to implement, robust, and adaptable to various model architectures.

Theory · Deep Learning

Hongkang Li, Hancheng Min, Rene Vidal

Transformer-based diffusion models have demonstrated remarkable performance at generating high-quality samples. However, our theoretical understanding of the reasons for this success remains limited. For instance, existing models are typically trained by minimizing a denoising objective, which is equivalent to fitting the score function of the training data. However, we do not know why transformer-based models can match the score function for denoising, or why gradient-based methods converge to the optimal denoising model despite the non-convex loss landscape. To the best of our knowledge, this paper provides the first convergence analysis for training transformer-based diffusion models. More specifically, we consider the population Denoising Diffusion Probabilistic Model (DDPM) objective for denoising data that follow a \textit{multi-token Gaussian mixture} distribution. We theoretically quantify the required number of tokens per data point and training iterations for the global convergence towards the Bayes optimal risk of the denoising objective, thereby achieving a desired score matching error. A deeper investigation reveals that the self-attention module of the trained transformer implements a \emph{mean denoising} mechanism that enables the trained model to approximate the oracle Minimum Mean Squared Error (MMSE) estimator of the injected noise in the diffusion steps. Numerical experiments validate these findings.

Probabilistic Methods · Gaussian Processes

Xiaoyu Jiang, Xinxing Shi, Sokratia Georgaka, Magnus Rattray, Mauricio Álvarez

Multi-Output Gaussian Processes (MOGPs) provide a principled probabilistic framework for modelling correlated outputs but face scalability bottlenecks when applied to datasets with high-dimensional output spaces. To maintain tractability, existing methods typically resort to restrictive assumptions, such as employing low-rank or sum-of-separable kernels, which can limit expressiveness. We propose the Transformed Latent Variable MOGP (T-LVMOGP), a novel framework that scales MOGPs to a massive number of outputs while preserving the capacity to capture meaningful inter-output dependencies. T-LVMOGP constructs a flexible multi-output deep kernel by mapping inputs and output-specific latent variables into an embedding space using a Lipschitz-regularised neural network. Combined with stochastic variational inference, our model effectively scales to high-dimensional output settings. Across diverse benchmarks, including climate modelling with over $10,000$ outputs and zero-inflated spatial transcriptomics data, T-LVMOGP outperforms baselines in both predictive accuracy and computational efficiency.

Deep Learning · Large Language Models

Jianbo Sun, Pengkun Yang

Hallucinations pose a key challenge for large language models. Chain-of-Thought prompting exposes intermediate reasoning, but reasoning traces are treated as linear traces, making it hard to capture cross-step dependencies and localize unsupported intermediate claims. We propose a \emph{structural reasoning model} to describe the interactions among local steps. To detect hallucinations, we extract a reasoning directed acyclic graph over conditions and intermediate claims, verify each claim against its parent nodes, and aggregate the step signals with a simple mass-flow rule. Under a probabilistic model, we provide an information-theoretic interpretation of this aggregation as measuring information loss along the reasoning graph. Experiments on GSM8K and MATH across multiple model families show that the proposed method improves detection performance over recent sampling-based baselines and judge-based methods. These findings provide a new perspective on the evaluation of chain-of-thought outputs and confirm the advantages of structural reasoning in hallucination detection.

Probabilistic Methods · Bayesian Models and Methods

Yuli Slavutsky, Ozgur Beker, David Blei, Bianca Dumitrascu

Disentangled representations separate factors that are shared across conditions from those that are condition-specific. Such separation is needed for generalization to new domains, treatments, patients, or species. A dominant line of work pursues this goal through variational formulations. While these approaches achieve partial disentanglement, they often exhibit three common limitations: they either do not remove all condition-specific information from the shared representation, allow the shared representation to become uninformative, or impose independence assumptions that do not reflect the underlying generative process. In this work, we introduce DisCoVR, a variational framework that addresses these limitations. Its objective is aligned with the probabilistic structure of the data-generating process, and includes an adversarial term that prevents condition-specific information from being encoded in the shared representation. DisCoVR reconstructs the data from both shared and condition-specific representations, ensuring that each remains informative, and uses a structured prior that further reinforces the informativeness of both representations. We show that across synthetic, image, and single-cell RNA-sequencing datasets, DisCoVR achieves stronger disentanglement compared to previous approaches.

Probabilistic Methods · Gaussian Processes

Viktoria Schram, Markus Hiller, Daniel Beck, Trevor Cohn

Predicting model performance at larger scales enables the design of training strategies and architectures tailored to specific performance targets. Empirical scaling law research identifies functional forms to aid this prediction task. These describe the relationship between loss and compute using a loss-compute frontier defined by learning curves. Due to the empirical nature of this approach, the computational burden is substantial, making strategic resource allocation essential -- yet it remains surprisingly underexplored. In this work, we address this shortcoming by exploring the suitability of Successive Halving (SH) and SH combined with parametric and non-parametric surrogate models. In addition to enabling a more systematic allocation of a given compute budget, our findings show that SH paired with surrogate models yields a set of learning curves that includes one with a lower loss-compute value than what naive uniform allocation or an SH-only approach can obtain. Our experiments demonstrate mean relative improvements of up to $2.84\%$ and $5.47\%$ on real-world and synthetic learning curve datasets. This strategic resource allocation enables us to obtain accurate scaling laws at significantly reduced computational costs, saving up to $98.7\%$ over the traditional exhaustive approach.