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Probabilistic Methods · Variational Inference

Mathis Chagneux, Mathias Müller, Pierre Gloaguen, Sylvain Le Corff, Jimmy Olsson

This article addresses online variational estimation in parametric state-space models. We propose a new procedure for efficiently computing the evidence lower bound and its gradient in a streaming-data setting, where observations arrive sequentially. The algorithm allows for the simultaneous training of the model parameters and the distribution of the latent states given the observations. It is based on i.i.d. Monte Carlo sampling, coupled with a well-chosen deep architecture, enabling both computational efficiency and flexibility. The performance of the method is illustrated on both synthetic data and real-world air-quality data. The proposed approach is theoretically motivated by the existence of an asymptotic contrast function and the ergodicity of the underlying Markov chain, and applies more generally to the computation of additive expectations under posterior distributions in state-space models.

Deep Learning · Generative Models and Autoencoders

Yuchen Jiao, Na Li, Changxiao Cai, Gen Li

Higher-order ODE solvers have become a standard tool for accelerating diffusion probabilistic model (DPM) sampling, motivating the widespread view that first-order methods are inherently slower and that increasing discretization order is the primary path to faster generation. This paper challenges this belief and revisits acceleration from a complementary angle: beyond solver order, the placement of DPM evaluations along the reverse-time dynamics can substantially affect sampling accuracy in the low-neural function evaluation (NFE) regime. We propose a novel training-free, first-order sampler named Forward DPMSolver (F-DPMSolver), whose leading discretization error has the opposite sign to that of DDIM. Algorithmically, the method approximates the forward-value evaluation via a cheap one-step lookahead predictor. We provide theoretical guarantees showing that the resulting sampler provably approximates the ideal forward-value trajectory while retaining first-order convergence. Empirically, across standard image generation benchmarks, the proposed sampler consistently improves sample quality under the same NFE budget and can be competitive with, and sometimes outperform, state-of-the-art higher-order samplers. Overall, the results suggest that the placement of DPM evaluations provides an additional and largely independent design angle for accelerating diffusion sampling. Our code is available at https://anonymous.4open.science/r/F-DPMSolver.

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.

Probabilistic Methods · Variational Inference

Yuhang Xi, Yu-Feng Yu, Chuan-Xian Ren, Zhao-Rong Lai

Parameter-Efficient Fine-Tuning (PEFT) is essential for adapting Large Language Models, yet existing methods typically struggle to balance model capacity with computational efficiency. Standard approaches often enforce rigid low-rank constraints, while dynamic alternatives incur significant memory overheads. To resolve this dilemma, we propose Spectral Bridge Variational Inference (SBVI), a geometric framework that reformulates LoRA not as static parameter optimization, but as a continuous Wasserstein gradient flow on the manifold of Gaussian measures. Rather than fixing the rank at initialization, SBVI governs the singular value evolution via a stochastic differential equation driven by a thermodynamic competition between task gradients and adaptive entropic friction. This mechanism induces a spectral bifurcation that automatically prunes redundant noise modes while amplifying signal-rich components, naturally discovering a layer-wise optimal rank distribution. We derive a scalable algorithm with linear complexity using factorized Riemannian retractions and an Empirical Bayes friction update. Experiments on reasoning and coding benchmarks demonstrate that SBVI achieves state-of-the-art performance, offering superior accuracy and memory efficiency compared to existing static and dynamic adaptation methods.

Probabilistic Methods · Monte Carlo and Sampling Methods

Aaron Havens, Brian Karrer, Neta Shaul

Sampling from unnormalized densities is analogous to the generative modeling problem, but the target distribution is defined by a known energy function instead of data samples. Evaluating the energy function is often costly, and thus a primary challenge is to learn an efficient sampler. We introduce *Flow Sampling*, a framework built on diffusion models and flow matching for the data-free setting. Our training objective is conditioned on a noise sample and regresses onto a *denoising* diffusion drift constructed from the energy function. In contrast, diffusion models' objective is conditioned on a data sample and regresses onto a *noising* diffusion drift. We utilize the interpolant process to minimize the number of energy function evaluations during training, resulting in an efficient and scalable method for sampling unnormalized densities. Furthermore, our formulation naturally extends to Riemannian manifolds, enabling diffusion-based sampling in geometries beyond the Euclidean space. We derive a closed-form formula for the conditional drift on constant curvature manifolds, including hyperspheres and hyperbolic spaces. We evaluate Flow Sampling on synthetic energy benchmarks, large-scale amortized molecular conformer generation, and distributions supported on the sphere, demonstrating strong empirical performance.

Probabilistic Methods · Monte Carlo and Sampling Methods

Robert Gruhlke, Julius Berner, David Sommer, Lorenz Richter

Diffusion models offer a powerful framework for sampling from complex probability densities by learning to reverse a noising process. A common approach involves solving for the time-reversed stochastic differential equation (SDE), which requires the score function of the evolving sample distribution. The logarithm of this distribution's density is governed by a Hamilton-Jacobi-Bellman (HJB) type partial differential equation (PDE). However, current methods for solving this PDE, such as PINNs or trajectory-based techniques, often suffer from long training times and significant sensitivity to hyperparameter tuning. In this work, we introduce a novel and efficient solver for the underlying HJB equation based on the functional tensor train (FTT) format. The FTT representation leverages latent low-rank structures to efficiently approximate high-dimensional functions, enabling both model compression and rapid computation. By integrating this efficient representation with a backward-in-time iterative scheme derived from backward stochastic differential equations (BSDEs), we develop a fast, robust and accurate sampling method. Our approach overcomes primary bottlenecks of existing techniques, enabling high-fidelity sampling from challenging target distributions with improved efficiency.

Probabilistic Methods · Bayesian Models and Methods

David Rügamer

Epistemic uncertainty is often viewed as a reducible uncertainty that vanishes with increasing data. This perspective implicitly assumes parameter identifiability and equates epistemic uncertainty with predictive variability. In overparametrized neural networks, however, model parameters are typically non-identifiable due to symmetries and redundant representations. As a consequence, substantial parameter uncertainty can persist even when the underlying function is fully identified. In this work, we analyze epistemic uncertainty through the lens of non-identifiability, characterize both discrete and continuous sources of residual uncertainty, and show that these can be measured using a variance-based decomposition. Focusing on one-hidden-layer ReLU networks, we thoroughly analyze the resulting posterior structure and validate our theoretical insights through empirical studies.

Probabilistic Methods · Bayesian Models and Methods

Albus Li, Matthew Wicker

Foundation models are increasingly being deployed in contexts where understanding the uncertainty of their outputs is critical to ensuring responsible deployment. While Bayesian methods offer a principled approach to uncertainty quantification, their computational overhead renders their use impractical for training or inference at foundation model scale. State-of-the-art models achieve parameter counts in the trillions through carefully engineered sparsity including Mixture-of-Experts (MoE) layers. In this work, we demonstrate calibrated uncertainty at scale by introducing Variational Mixture-of-Experts Routing (VMoER), a structured Bayesian approach for modelling uncertainty in MoE layers. VMoER confines Bayesian inference to the expert-selection stage which is typically done by a deterministic routing network. We instantiate VMoER using two inference strategies: amortised variational inference over routing logits and inferring a temperature parameter for stochastic expert selection. Across tested foundation models, VMoER improves routing stability under noise by 38%, reduces calibration error by 94%, and increases out-of-distribution AUROC by 12%, while incurring less than 1% additional FLOPs. These results suggest VMoER offers a scalable path toward robust and uncertainty-aware foundation models.

Probabilistic Methods · Graphical Models

Leander Kurscheidt, Gabriele Masina, Roberto Sebastiani, Antonio Vergari

In many safety-critical settings, probabilistic ML systems have to make predictions subject to algebraic constraints, e.g., predicting the most likely trajectory that does not cross obstacles. These real-world constraints are rarely convex, nor the densities considered are (log-)concave. This makes computing this constrained maximum a posteriori (MAP) prediction in an efficient and reliable way extremely challenging. In this paper, we first investigate under which conditions we can perform constrained MAP inference over continuous variables exactly and efficiently and devise a scalable message-passing algorithm for this tractable fragment. Then, we devise a general constrained MAP strategy that interleaves partitioning the domain into convex feasible regions with numerical constrained optimization. We evaluate both methods on synthetic and real-world benchmarks, showing our structure aware approach outperforms constraint-agnostic baselines.

Theory · Probabilistic Methods

Weihao Li, Jun Liu

Logistic regression is widely used in applications; however, when the dimension scales with the sample size, theory reveals that the asymptotic behavior of common M-estimators depends on bias and variance scaling constants, which are functions of the signal strength. To leverage the theory to design statistical methodologies, it is essential to obtain accurate estimates of the signal strength. In this work, we utilize a data-splitting strategy to efficiently estimate the signal strength. To alleviate issues caused by separable data, we analyze the exact asymptotics of an M-estimator with a data-driven, non-decomposable regularizer that adapts to the true covariance structure. We justify the validity of our method through both theoretical analysis and numerical experiments.

Haowen Li, Tianxiang Li, Yi Yang, Boyu Cao, Qi Liu

The advancement of diffusion-based text-to-music generation has opened new avenues for zero-shot music editing. However, existing methods fail to achieve context-aware editing, which requires altering specific stems while strictly preserving the background accompaniment. This limitation severely hinders practical application, since real-world production necessitates precise manipulation of components within dense mixtures. Our key finding is that, while vanilla cross-attention captures semantic intent, it lacks the spectral resolution to strictly localize targets in dense mixtures, leading to boundary leakage. To resolve this dilemma, we propose Polyphonia, a training-free editing framework with Acoustic-Informed Attention Calibration. Rather than relying solely on diffuse semantic attention, Polyphonia leverages a probabilistic acoustic prior to establish coarse boundaries, enabling background context preserved precise semantic synthesis. For evaluation, we propose PolyEvalPrompts, a standardized prompt set with 1,170 context-aware music editing tasks. Specifically, Polyphonia achieves an increase of 15.5% in target alignment compared to baselines, while maintaining competitive music fidelity and background integrity.

Reinforcement Learning · Deep RL

Youngin Kim, Ray Sun, Inho Kim, Bumsoo Park, Hyun Oh Song

Transformer-based world models have shown strong performance in visual reinforcement learning, but often suffer from temporal inconsistency in long-horizon rollouts, including object duplication, disappearance, and transmutation. A key reason is that most existing approaches treat next-frame prediction purely as a token generation problem, without explicitly modeling correspondence between tokens across time. We formulate next-frame prediction as a structured probabilistic inference problem with latent token correspondence variables, deriving a model in which each next-frame token is explained either by copying a token from the previous frame or by generating a new token. Our experiments show state-of-the-art performance on 4 challenging benchmarks. The proposed method achieves a return of 72.5% and a score of 35.6% on the Craftax-classic benchmark, significantly surpassing the previous best of 67.4% and 27.9%. We plan to release our source code on GitHub upon acceptance.

Deep Learning · Large Language Models

Rachel Lawrence, Jacqueline Maasch

Autonomous reasoning is among the most scientifically and economically motivating topics in AI today. Historically the purview of symbolic AI, recent advances have mainly emerged from deep probabilistic generative models. Despite immense interest and rapid progress, the generative AI community has not clearly converged on operational definitions for reasoning and often implicitly rejects the historical treatment of this topic in logic, verifiable automated reasoning, and symbolic methods in general. **This position contends that definitional ambiguity leaves the construct validity of reasoning evaluation unverifiable, and undermines quantifiable progress toward the collective goal of trustworthy autonomous reasoning.** We also contend that this ambiguity is addressable. To that end, we provide (1) general and extensible definitions for *valid* and *sound reasoning* based on a synthesis of the literature, which can serve as an accessible reference and a starting point for community discussion; and (2) a checklist for best practices in the communication of AI reasoning research.

Theory · Probabilistic Methods

Qiaoyu Liang, Haohua Chen, Zihan Zhu, Michael Evans

From a statistical evidence perspective, we establish some asymptotic optimality properties of certain multiple testing rules based on the relative belief ratio (Evans, 2015). Under the two-groups model with an additive 0-1 loss and within a Bayesian decision theoretic asymptotic framework of Bogdan et al. (2011), we show that relative belief multiple testing rules induced by a simple one-group light-tailed normal prior with a single hyperparameter achieve the same asymptotic Bayes risk as the Bayes oracle benchmark. This risk is the minimum achievable in this asymptotic framework. Despite originating from a different starting point, the evidential relative belief approach enjoys oracle properties. The relative belief multiple testing approach is fundamentally different from existing Bayesian multiple testing procedures, virtually all induced by more complex heavy-tailed one-group global-local shrinkage priors using purely posterior-based inferences (Datta & Ghosh, 2013; Ghosh et al., 2016; Bhadra et al., 2017; Ghosh & Chakrabarti, 2017; Qin & Ghosh, 2025). By measuring statistical evidence via both the prior and posterior, the relative belief approach reveals an alternative new inferential paradigm for attaining asymptotic Bayes optimality under sparsity, one that does not rely on developing increasingly elaborate priors.

Probabilistic Methods · Monte Carlo and Sampling Methods

Beatrice Franzolini, Francesco Gaffi

Slice sampling is a standard Monte Carlo technique for Dirichlet process (DP)--based models, widely used in posterior simulation. However, formal assessments of the scalability of posterior slice samplers have remained largely unexplored, primarily because the computational cost of a slice-sampling iteration is random and potentially unbounded. In this work, we obtain high-probability bounds on the computational complexity of DP slice samplers. Our main results show that, uniformly across posterior cluster-growth regimes, the overhead induced by slice variables, relatively to the number of clusters supported by the posterior, is $O_{\mathbb P}(\log n)$. As a consequence, even in worst-case configurations, superlinear blow-ups in per-iteration computational cost occur with vanishing probability. Our analysis applies broadly to DP--based models without any likelihood-specific assumptions, still providing complexity guarantees for posterior sampling on arbitrary datasets. These results establish a theoretical foundation for assessing the practical scalability of slice sampling in DP-based models.

Probabilistic Methods · Everything Else

Anton Conrad, Eric Moulines, julien perez

In multi-target regression and multi-class classification, uncertainty is inherently multivariate: prediction regions must capture joint dependencies across correlated outputs. Conformal prediction provides distribution-free guarantees, yet extending it to vector-valued outputs remains challenging—scalar aggregation discards geometric structure, while optimal transport (OT) approaches are computationally demanding and sensitive to outliers. We introduce two conformal methods based on geometric quantiles and spatial ranks: Geometric Conformalized Quantile Regression (GCQR) constructs prediction regions from learned conditional geometric quantiles, while Geometric Rank Conformal Prediction (GRCP) uses the radial rank of vector-valued conformity scores as the nonconformity measure. We propose multiple estimators offering different tradeoffs between computational cost and adaptivity to feature-dependent heterogeneity, with scalable learning via partially input-convex neural networks. On multi-target regression and multi-class classification benchmarks, GCQR and GRCP attain near-nominal coverage with consistently tighter prediction regions than scalarized and multivariate baselines.

Probabilistic Methods · Graphical Models

Samo Hromadka, Kai Biegun, Lior Fox, James Heald, Maneesh Sahani

We address the challenge of uncovering systematic, and potentially controllable, dynamical structure underlying complex high-dimensional time series data. Existing generative and autoregressive models have difficulty telling systematic content apart from distractors, while contrastive methods struggle to learn accurate dynamics. To address these shortcomings, we develop the Recognition-Parametrized Gaussian State Space Model (RP-GSSM), a probabilistic framework that infers accurate latent dynamics without relying on a parametrized decoder. By eliminating explicit generative parameters, the model directs its entire representational capacity to encoding dynamically relevant state; and, being fully probabilistic, learns via maximum likelihood without auxiliary objectives or ad-hoc regularization. Combining the expressive power of a neural network encoder with exact inference under a jointly Gaussian prior allows the RP-GSSM to embed a broad class of intrinsically nonlinear dynamical systems. We show that the RP-GSSM recovers physically meaningful latent states from noisy video more faithfully than competing methods, more reliably identifies underlying controllable nonlinear dynamics, and remains substantially more robust to visual distractors.

Optimization · Discrete and Combinatorial Optimization

Stefan Clarke, Bartolomeo Stellato

Mixed-integer optimization solvers often find optimal solutions early in the search, yet spend the majority of computation time proving optimality. We exploit this by learning when to terminate solvers early on distributions of similar problem instances. Our method trains a neural network to estimate the true optimality gap from the solver state, then uses conformal prediction to calibrate a stopping threshold with rigorous probabilistic guarantees on solution quality. On five problem families from the distributional MIPLIB library, our method reduces solve time by over 60% while guaranteeing 0.1%-optimal solutions with 95% probability.

Probabilistic Methods · Bayesian Models and Methods

Yusuke Sakai, Hidetaka Kamigaito, Taro Watanabe

Minimum Bayes Risk (MBR) decoding yields more robust and higher-quality text generation than maximum a posteriori (MAP) decoding by selecting hypotheses that maximize expected utility over sampled pseudo-references. However, there exists discrepancy in the design: hypothesis selection calculates expected utility scores conditioned on given pseudo-references, while commonly used evaluation metrics, e.g., BLEU and COMET, are asymmetric. Therefore, it is important to consider both hypothesis-to-reference and reference-to-hypothesis directional effects. In this study, we introduce a noisy channel decomposition of MBR decoding that naturally incorporates bidirectional effects to account for these asymmetries. We decompose MBR decoding into four interacting components: hypothesis-to-reference likelihood, reference-to-hypothesis likelihood, hypothesis prior, and reference prior. This decomposition provides a unified interpretation of existing MBR variants and enables metric- and task-specific interpretability by isolating the contribution of each channel. Furthermore, our comprehensive analysis demonstrates that appropriate channel weighting consistently yields performance gains over original MBR decoding across tasks and utility functions.