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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.

Applications · Computer Vision

Yin Tang, Jiawei Ma, Jinrui Zhang, Alex Jinpeng Wang, Deyu Zhang

Continuous prediction in complex environments is critical for Unmanned Aerial Vehicle (UAV). However, the existing Vision-Language Navigation (VLN) models follows the dead-reckoning, which iteratively predicts the next waypoint and updates its position, thereby constructing the complete trajectory. Then, such stepwise manner will inevitably lead to accumulated errors of position over time, resulting in misalignment between internal belief and objective coordinates, which is known as ``state drift'' and ultimately compromises the subsequent trajectory prediction. Drawing inspiration from classical control theory, we propose to correct for errors by formulating the continuous prediction as a recursive Bayesian state estimation problem. In this paper, we design NeuroKalman, a novel framework that decouples navigation into two complementary processes: a Prior Prediction, based on motion dynamic,s and a Likelihood Correction, from historical observation. We first mathematically associate Kernel Density Estimation of the measurement likelihood with the attention-based retrieval mechanism, which then allows the system to rectify the latent representation using retrieved historical anchors without gradient updates. Comprehensive experiments on TravelUAV benchmark demonstrate that, with only 10\% of the full training data fine-tuning, our method clearly outperforms strong baselines and regulates drift accumulation.

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.

Applications · Language, Speech and Dialog

Mengyi DENG, Zhiwei Li, Xin Li, Tingyu ZHU, Ying Zhao, Zhijiang Guo, Wei Wang

Large Language Model (LLM) agents often operate under underspecified user instructions, where latent uncertainty over user intent leads to erroneous tool actions. To address this challenge, we propose a goal-oriented clarification framework that aligns clarification behavior with ambiguity resolution. Central to our approach is the Information Gain Reward, a metric that quantifies the utility of clarification questions by measuring the Bayesian belief update towards the ground-truth goal induced by the clarification exchange. We train the clarifier (LLM) using this reward to optimize for high information gain, ensuring that clarifications effectively reduce uncertainty and improve task completion within the agent-tool-user environment. We validate our framework within a clarification-enhanced $\tau$-Bench environment, conducting cross-agent evaluations across five heterogeneous backbones. Empirical results demonstrate that our method consistently improves the success rate by 3.7\% over the no-clarification baseline, incurring small overhead via surgical interventions (averaging only 0.3 additional steps). Our code and data are available at the anonymous repository: \url{https://anonymous.4open.science/r/Information-Gain-Clarifier-AD68/}.

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.

Probabilistic Methods · Bayesian Models and Methods

Zier Mensch, Lars Holdijk, Samuel Duffield, Maxwell Aifer, Patrick Coles, Max Welling, Miranda C. N. Cheng

Stochastic-gradient MCMC methods enable scalable Bayesian posterior sampling but often suffer from sensitivity to minibatch size and gradient noise. To address this, we propose Stochastic Gradient Lattice Random Walk (SGLRW), an extension of the Lattice Random Walk discretization. Unlike conventional Stochastic Gradient Langevin Dynamics (SGLD), SGLRW introduces stochastic noise only through the off-diagonal elements of the update covariance; this yields greater robustness to minibatch size while retaining asymptotic correctness. Furthermore, as comparison we analyze a natural analogue of SGLD utilizing gradient clipping. Experimental validation on Bayesian regression and classification demonstrates that SGLRW remains stable in regimes where SGLD fails, including in the presence of heavy-tailed gradient noise, and matches or improves predictive performance.

Probabilistic Methods · Bayesian Models and Methods

Kaizheng Wang, Yuhang Wu, Assaf Zeevi

We study adaptive querying for learning user-dependent quantities of interest, such as responses to held-out items and psychometric indicators, within tight question budgets. Classical Bayesian design and computerized adaptive testing typically rely on restrictive parametric assumptions or expensive posterior approximations, limiting their use in heterogeneous, high-dimensional, and cold-start settings. We introduce a persona-induced latent variable model that represents a user's state through membership in a finite dictionary of AI personas, each offering response distributions produced by a large language model. This yields expressive priors with closed-form posterior updates and efficient finite-mixture predictions, enabling scalable Bayesian design for sequential item selection. Experiments on synthetic data and WorldValuesBench demonstrate that persona-based posteriors deliver accurate probabilistic predictions and an interpretable adaptive elicitation pipeline.

Vasily Ilin, Peter Sushko, Ranjay Krishna

Estimating probability density and its score from samples remains a core problem in generative modeling, Bayesian inference, and kinetic theory. Existing methods are bifurcated: classical kernel density estimators (KDE) generalize across distributions but suffer from the curse of dimensionality, while modern neural score models achieve high precision but require retraining for every target distribution. We introduce DiScoFormer (Density and Score Transformer), a ``train-once, infer-anywhere" equivariant Transformer that maps i.i.d. samples to both density values and score vectors, generalizing across distributions and sample sizes. Analytically, we prove that self-attention can recover normalized KDE, establishing it as a functional generalization of kernel methods; empirically, individual attention heads learn multi-scale, kernel-like behaviors. The model converges faster and achieves higher precision than KDE for density estimation, and provides a high-fidelity plug-in score oracle for score-debiased KDE, Fisher information computation, and Fokker-Planck-type PDEs.

General Machine Learning · Kernel methods

Vasily Ilin, Peter Sushko, Ranjay Krishna

Estimating probability density and its score from samples remains a core problem in generative modeling, Bayesian inference, and kinetic theory. Existing methods are bifurcated: classical kernel density estimators (KDE) generalize across distributions but suffer from the curse of dimensionality, while modern neural score models achieve high precision but require retraining for every target distribution. We introduce DiScoFormer (Density and Score Transformer), a ``train-once, infer-anywhere" equivariant Transformer that maps i.i.d. samples to both density values and score vectors, generalizing across distributions and sample sizes. Analytically, we prove that self-attention can recover normalized KDE, establishing it as a functional generalization of kernel methods; empirically, individual attention heads learn multi-scale, kernel-like behaviors. The model converges faster and achieves higher precision than KDE for density estimation, and provides a high-fidelity plug-in score oracle for score-debiased KDE, Fisher information computation, and Fokker-Planck-type PDEs.

Applications · Computer Vision

Saman Forouzandeh, Wei Peng, Xinghuo Yu, Mahdi Jalili

VideoQA models can be accurate yet often fail to align answers with the correct video segments (the \emph{grounding gap}). We introduce \textbf{LINGUA} (\textbf{L}anguage-based \textbf{IN}ference for \textbf{G}rounded Video \textbf{U}nderstanding \textbf{A}gent), a memory-based agent that performs grounded VideoQA by reasoning in an explicit \emph{linguistic belief state}. LINGUA uses five mechanisms: (1) event-driven perception (retains 8--12\% of frames while preserving 94\% of question-relevant events); (2) typed memory for episodic narratives, semantic affordances, and procedural scripts; (3) Belief-Action-Verification loops with postcondition and temporal checks; (4) meta reflection with contrastive refinement; and (5) Bayesian reliability tracking for continual learning without gradient updates. Built with Gemma3-4B (Ollama, 4-bit), LINGUA outperforms strong baselines on five VideoQA benchmarks, reaching 82.4\% on NExT-QA and 42.3\% Acc@GQA on NExT-GQA (answer + IoU$\geq$0.5 temporal localization), while running 2.6$\times$ faster than dense-frame methods. In continual learning over 100 videos, accuracy rises from 45.2\% (first 10) to 61.8\% (last 10) without catastrophic forgetting, indicating online adaptation via memory refinement.

Probabilistic Methods · Bayesian Models and Methods

Horace Yiu, Leandro Sánchez-Betancourt, Alvaro Cartea, Gerardo Duran-Martin

We derive a robust update rule for the online infinite hidden Markov model (iHMM) for when the streaming data contains outliers and the model is misspecified. Leveraging recent advances in generalised Bayesian inference, we define robustness via the posterior influence function (PIF), and provide conditions under which the online iHMM has bounded PIF. Imposing robustness inevitably induces an adaptation lag for regime switching. Our method, which is called Batched Robust iHMM (BR-iHMM), balances adaptivity and robustness with two additional tunable parameters. Across limit order book data, hourly electricity demand, and a synthetic high-dimensional linear system, BR-iHMM reduces one-step-ahead forecasting error by up to 67% relative to competing online Bayesian methods. Together with theoretical guarantees of bounded PIF, our results highlight the practicality of our approach for both forecasting and interpretable online learning.

Applications · Robotics

Yixian Chen, Rufan Bai, Jiangbin Zheng, Yimin Wang, Tiantian CHEN, Wei Wang, Yuhuan Lu

Autonomous vehicles operating in open-world environments must continually adapt to rare long-tail scenarios while preserving previously acquired driving skills. However, existing trajectory planning approaches struggle with this stability-plasticity trade-off, as they rely on static models or rigid rule-based controllers that cannot robustly handle evolving and complex traffic dynamics. Against this background, we propose **NOMAD**, a lifelong trajectory planning framework that integrates non-parametric Bayesian memory with diffusion-based trajectory generation, enabling continuous adaptation to long-tail scenarios without catastrophic forgetting. Our method maps continuous scene contexts to a dynamically growing set of discrete memory clusters, which guide a conditional diffusion model to function as a mixture of experts specialized for diverse driving behaviors. To retain past knowledge during incremental learning, we introduce a generative replay mechanism that synthesizes pseudo-experiences from previously learned memory clusters. Extensive closed-loop evaluations on the nuPlan benchmark demonstrate that our approach achieves state-of-the-art performance on long-tail scenarios, improving the interPlan score by **9.4\%** over the strongest baseline, while maintaining competitive performance on regular driving benchmarks. Moreover, our method exhibits robust continual learning capability, achieving the highest average closed-loop score with positive backward transfer when adapting to sequentially introduced long-tail scenarios.

Probabilistic Methods · Bayesian Models and Methods

Kenyon Ng, Edwin Fong, David Frazier, Jeremias Knoblauch, Susan Wei

Bayesian inference provides principled uncertainty quantification but is often limited by challenges of prior and likelihood elicitation. The martingale posterior (MGP) (Fong et al, 2023) offers an alternative by replacing these requirements with a predictive rule. Additionally MGP focuses inference on parameters defined through a loss function. This framework is especially resonant in the era of foundation transformers; practitioners increasingly leverage models like TabPFN for their state-of-the-art capabilities, yet often require epistemic uncertainty for a scientific estimand $\theta$ that need not parametrize the model’s implicit latent model. The MGP provides the mechanism to recover these posterior distributions. We introduce TabMGP, an MGP built on TabPFN for tabular data. TabMGP produces credible sets with near-nominal coverage and often outperforms both handcrafted MGP constructions and standard Bayesian baselines.

Deep Learning · Large Language Models

Yun Qu, Qi Wang, Yixiu Mao, Heming Zou, Yuhang Jiang, Weijie Liu, Clive Bai, Kai Yang, Yangkun Chen, Saiyong Yang 等

Reinforcement learning enhances the reasoning capabilities of large language models but often involves high computational costs due to rollout-intensive optimization. Online prompt selection presents a plausible solution by prioritizing informative prompts to improve training efficiency. However, current methods either depend on costly, exact evaluations or construct prompt-specific predictive models lacking generalization across prompts. This study introduces Generalizable Predictive Prompt Selection (GPS), which performs Bayesian inference towards prompt difficulty using a lightweight generative model trained on the shared optimization history. Intermediate-difficulty prioritization and history-anchored diversity are incorporated into the batch acquisition principle to select informative prompt batches. The small predictive model also generalizes at test-time for efficient computational allocation. Experiments across varied reasoning benchmarks indicate GPS's substantial improvements in training efficiency, final performance, and test-time efficiency over superior baseline methods. The code is available at https://anonymous.4open.science/r/GPS-ICML.

Probabilistic Methods · Monte Carlo and Sampling Methods

Xiaoyu Wang, Jonathan Huggins

Stochastic gradient Langevin dynamics combined with Gibbs updates (SGLD-Gibbs) provides a highly scalable approach to approximate Bayesian inference in latent variable models. However, it remains unclear how to tune the algorithm's hyperparameters in a principled manner to ensure the uncertainty estimates are statistically meaningful. In this work, we address this gap in tuning guidance by developing a statistical scaling limit theory for SGLD-Gibbs. We derive a joint asymptotic limit for the global parameters and latent variables under appropriate space-time rescaling. We show that global parameters converge to a diffusion-type limit, while individual latent variables converge to a jump process reflecting their intermittent Gibbs updates. This joint jump-diffusion structure reveals how latent-variable randomness contributes to the stationary distribution of the global parameters. We leverage our results to provide explicit guidance on hyperparameter tuning for SGLD-Gibbs that ensures meaningful uncertainty quantification. Our empirical results show that SGLD-Gibbs with our tuning guidance leads to better parameter estimates and uncertainty quantification than stochastic variational inference.

Rachael Hwee Ling Sim, Jue Fan, Xiao Tian, Xinyi Xu, Patrick Jaillet, Bryan Kian Hsiang Low

Collaborative machine learning involves training high-quality models using datasets from a number of sources. To incentivize sources to share data, existing data valuation methods fairly reward each source based on its data submitted as is. However, as these methods do not verify nor incentivize data truthfulness, the sources can manipulate their data (e.g., by submitting duplicated or noisy data) to artificially increase their valuations and rewards or prevent others from benefiting. This paper presents the first mechanism that provably ensures (**F**) collaborative fairness and incentivizes (**T**) truthfulness at equilibrium for Bayesian models. Our mechanism combines semivalues (e.g., Shapley value), which ensure fairness, and a truthful data valuation function (DVF) based on a validation set that is unknown to the sources. As semivalues are influenced by others' data, we introduce an additional condition to prove that a source can maximize its expected data values in coalitions and semivalues by submitting a dataset that captures its true knowledge. Additionally, we discuss the implications and suitable relaxations of (**F**) and (**T**) when the mediator has a limited budget for rewards or lacks a validation set. Our theoretical findings are validated on synthetic and real-world datasets.

Probabilistic Methods · Variational Inference

Sylvia Liu, Valeria Vitelli, Carlo Mannino, Arnoldo Frigessi, Ida Scheel

We propose the Pseudo-Mallows distribution over the set of all permutations of $n$ items, to approximate the posterior distribution of the Bayesian Mallows model. The Bayesian Mallows model has been successfully used for recommender systems to learn personal preferences from highly incomplete users data. However current inference algorithms do not scale, preventing its use in real-time applications. The Pseudo-Mallows distribution is a product of univariate discrete Mallows-like distributions, where the quality of the approximation depends on the order of the $n$ items in the factorization sequence. In a variational setting, we optimize the variational order parameter by minimising a marginalized KL-divergence, conjecturing a certain form of the optimal variational order that depends on the data, and proposing an approximation algorithm for this discrete optimization. Empirical evidence and some theory support our conjecture. We demonstrate on clicking data that variational inference via the Pseudo-Mallows distribution allows much faster probabilistic preference learning compared to alternative MCMC-based options.

Deep Learning · Large Language Models

Changhai Zhou, Shiyang Zhang, Yuhua Zhou, Qian Qiao, Jun Gao, Cheng Jin, KAIZHOU QIN, Weizhong Zhang

Quantization followed by parameter-efficient fine-tuning has emerged as a promising paradigm for downstream adaptation under tight GPU memory constraints. However, this sequential pipeline fails to leverage the intricate interaction between quantization bit-width and LoRA rank. Specifically, a carefully optimized quantization allocation with low quantization error does not always translate to strong fine-tuning performance, and different bit-width and rank configurations can lead to significantly varying outcomes under the same memory budget. To address this limitation, we propose AutoQRA, a joint optimization framework that simultaneously optimizes the bit-width and LoRA rank configuration for each layer during the mixed quantized fine-tuning process. To tackle the challenges posed by the large discrete search space and the high evaluation cost associated with frequent fine-tuning iterations, AutoQRA decomposes the optimization process into two stages. First, it first conducts a global multi-fidelity evolutionary search, where the initial population is warm-started by injecting layer-wise importance priors. This stage employs specific operators and a performance model to efficiently screen candidate configurations. Second, trust-region Bayesian optimization is applied to locally refine promising regions of the search space and identify optimal configurations under the given memory budget. This approach enables active compensation for quantization noise in specific layers during training. Experiments show that AutoQRA achieves performance close to full-precision fine-tuning with a memory footprint comparable to uniform 4-bit methods.

Optimization · Zero-order and Black-box Optimization

Baek Seong-Eun, Lee Jung-Mok, Kim Sung-Bin, Tae-Hyun Oh

Fine-tuning Large Language Models (LLMs) with Low-Rank Adaptation (LoRA) offers a resource-efficient way to personalize or specialize. However, LoRA is highly sensitive to hyperparameter choices, and performing an exhaustive hyperparameter search remains computationally intensive. To address these challenges, we propose a framework that integrates the domain knowledge of pre-trained LLMs into the Bayesian Optimization (BO) process to efficiently search for LoRA hyperparameters. To leverage pre-trained LLMs' knowledge, our approach repurposes them as a discrete-to-continuous mapping module to link hyperparameters and their domain knowledge to a continuous vector space, where BO is conducted. We design and control the mapping via language prompting, providing a domain-aware textual prompt that describes the relationships among hyperparameters and their respective roles. This allows us to explicitly inject domain knowledge about LoRA into the LLM in natural language. We also introduce an additional learnable token to capture residual information that is difficult to describe linguistically in the prompt. This aids BO to sample more high-performing hyperparameters. In addition, by leveraging the strong correlation observed between the performance obtained from full and subset training datasets in LoRA training regimes, we introduce proxy training and evaluation using a data subset. This significantly improves the efficiency of our method. We demonstrate that our hyperparameter, discovered with only about 30 iterations, achieves more than 20% performance improvement over standard hyperparameters found from about 45,000 combinations. Code will be released.