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Probabilistic Methods · Monte Carlo and Sampling Methods

Samuel Howard, Nikolas Nüsken, Jakiw Pidstrigach

Simulating the conditioned dynamics of diffusion processes, given their initial and terminal states, is an important but challenging problem in the sciences. The difficulty is particularly pronounced for rare events, for which the unconditioned dynamics rarely reach the terminal state. In this work, we propose a novel approach for learning diffusion bridges based on a self-consistency property of the optimal control. The resulting algorithm learns the conditioned dynamics in an iterative online manner, and exhibits strong performance in a range of empirical settings without requiring differentiation through simulated trajectories. Beyond the diffusion bridge setting, we draw connections between our self-consistency framework and recent advances in the wider stochastic optimal control literature.

Probabilistic Methods · Everything Else

Meshi Bashari, Yonghoon Lee, Roy Lotan, Edgar Dobriban, Yaniv Romano

The rapid proliferation of high-quality synthetic data---generated by advanced AI models or collected as auxiliary data from related tasks---presents both opportunities and challenges for statistical inference. This paper introduces a GEneral Synthetic-Powered Inference (GESPI) framework that wraps around any statistical inference procedure to safely enhance sample efficiency by combining synthetic and real data. Our framework leverages high-quality synthetic data to boost statistical power, yet adaptively defaults to the standard method using only real data when synthetic data are of low quality. The error rate of our method remains below a user-specified bound without any distributional assumptions on the synthetic data, and decreases as the quality of the synthetic data improves. This flexibility enables seamless integration with conformal prediction, risk control, hypothesis testing, and multiple testing procedures, all without modifying the base inference method. We demonstrate the benefits of our method on challenging tasks with limited labeled data, including AlphaFold protein structure prediction, and comparing large reasoning models on complex math problems.

Probabilistic Methods · Everything Else

Shalev Shaer, Yarin Bar, Drew Prinster, Yaniv Romano

We propose a sequential test for distribution-shift detection that allows conformal test martingales (CTMs) to work under a fixed, reference-conditional setting. Existing CTM detectors construct test martingales by continually growing a reference set with each incoming sample, using it to assess how atypical the new sample is relative to past observations. While this design yields anytime-valid type-I error control, it suffers from test-time contamination: after a change, post-shift observations enter the reference set and dilute the evidence for distribution shift, increasing detection delay and reducing power. In contrast, our method avoids contamination by design by comparing each new sample to a fixed null reference dataset. Our main technical contribution is a robust martingale construction that remains valid conditional on the null reference data, achieved by explicitly accounting for the estimation error in the reference distribution induced by the finite reference set. This yields anytime-valid type-I error control together with guarantees of asymptotic power one and bounded expected detection delay. Empirically, our method detects shifts faster than standard CTMs, providing a powerful and reliable distribution-shift detector.

Probabilistic Methods · Spectral Methods

Roddy Taing, Keith Levin

As network data has become ubiquitous in the sciences, there has been growing interest in network models whose structure is driven by latent node-level variables in a (typically low-dimensional) latent geometric space. These "latent positions" are often estimated via embeddings, whereby the nodes of a network are mapped to points in Euclidean space so that "similar" nodes are mapped to nearby points. Under certain model assumptions, these embeddings are consistent estimates of the latent positions, but most such results require the embedding dimension to be chosen correctly. Methods for choosing the embedding dimension have been studied extensively, but little is known about the behavior of embeddings when the dimension is misspecified. In this work, we provide a theoretical description of the effects of dimension misspecification under the random dot product graph, a class of latent space network models that includes several widely-used network models, most notably the stochastic blockmodel, as special cases. We show that when the dimension is chosen too large, consistent estimation still holds, albeit at a slower rate than when the embedding dimension is chosen correctly. On the other hand, when the dimension is chosen too small, there is a fundamental estimation error lower bound that need not go to zero in the large-network limit. A range of synthetic data experiments support our theoretical results. Our main technical result, which may be of independent interest, is a generalization of earlier work in random matrix theory showing that all non-signal eigenvectors of a low-rank matrix subject to additive noise are delocalized.

Deep Learning · Large Language Models

Yang Yang, Hua XU, Zhangyi Hu, Yutao Yue

Large Language Models (LLMs) can propose natural-language rules, circumventing the reliance on a predefined predicate space in traditional rule learning. However, existing LLM-based methods often neglect the global interactions among rules, and the potential of using fine-grained rule importance scores to calibrate neuro-symbolic reasoning remains underexplored. To address this gap, we introduce RLIE, a framework that integrates LLMs with probabilistic modeling to learn weighted rule sets in four stages: (1) Rule generation: proposing and filtering candidate rules via LLMs; (2) Logistic regression: learning sparse, calibrated weights for global rule selection; (3) \textbf{I}terative refinement: revising the rule set with error-driven hard examples; and (4) \textbf{E}valuation: {validating the learned system via comparative inference paradigms}. Across multiple real-world datasets and LLM backbones, our learned weighted rules \textbf{achieve superior stability and accuracy}, whereas rule-injection prompting yields mixed results and often degrades performance. These results suggest LLMs excel at semantic rule discovery but are less reliable at controlled probabilistic aggregation. Our findings highlight both the promise and the limits of LLMs for inductive reasoning, motivating a principled integration with classic probabilistic rule combination for reliable neuro-symbolic reasoning.

Probabilistic Methods · Bayesian Models and Methods

Dongqing Li, Zheqiao Cheng, Geoff Nicholls, Quyu Kong

AI agents increasingly execute procedural workflows as sequential action traces, which obscures latent concurrency and induces repeated step-by-step reasoning. We introduce BPOP, a Bayesian framework that infers a latent dependency partial order from noisy linearized traces. BPOP models traces as stochastic linear extensions of an underlying graph and performs efficient MCMC inference via a tractable frontier-softmax likelihood that avoids \#P-hard marginalization over linear extensions. We evaluate on our open-sourced Cloud-IaC-6, a suite of cloud provisioning tasks with heterogeneous LLM-generated traces, and WFCommons scientific workflows. BPOP recovers dependency structure more accurately than trace-only and process-mining baselines, and the inferred graphs support a compiled executor that prunes irrelevant context, yielding substantial reductions in token usage and execution time.

Probabilistic Methods · Everything Else

George Whittle, Juliusz Ziomek, Jacob Rawling, Michael A Osborne

While Bayesian inference provides a principled framework for reasoning under uncertainty, its widespread adoption is limited by the intractability of exact posterior computation, necessitating the use of approximate inference. However, existing methods are often computationally expensive, or demand costly retraining when priors change, limiting their utility, particularly in sequential inference problems such as real-time sensor fusion. To address these challenges, we introduce the Distribution Transformer---a novel architecture that can learn arbitrary distribution-to-distribution mappings. Our method can be trained to map a prior to the corresponding posterior, conditioned on some dataset---thus performing approximate Bayesian inference. Our novel architecture represents a prior distribution as a (universally-approximating) Gaussian Mixture Model (GMM), and transforms it into a GMM representation of the posterior. The components of the GMM attend to each other via self-attention, and to the datapoints via cross-attention. We demonstrate that Distribution Transformers both maintain flexibility to vary the prior, and significantly reduces computation times—from minutes to milliseconds—while achieving expected log-likelihood performance on par with or superior to existing approximate inference methods across tasks such as sequential inference, quantum system parameter inference, and Gaussian Process predictive posterior inference with hyperpriors.

Probabilistic Methods · Bayesian Models and Methods

Rajeev Yasarla, Shizhong Han, Hsin-Pai Cheng, Apratim Bhattacharyya, Shweta Mahajan, Litian Liu, Yunxiao Shi, Risheek Garrepalli, Hong Cai, Fatih Porikli

End-to-end (E2E) autonomous driving has recently emerged as a new paradigm, offering significant potential. However, few studies have looked into the practical challenge of deployment across domains (e.g., cities). Although several works have incorporated Large Language Models (LLMs) to leverage their open-world knowledge, LLMs do not guarantee cross-domain driving performance and may incur prohibitive retraining costs during domain adaptation. In this paper, we propose RoCA, a novel framework for robust cross-domain E2E autonomous driving. RoCA formulates the joint probabilistic distribution over the tokens that encode ego and surrounding vehicle information in the E2E pipeline. Instantiating with a Gaussian process (GP), RoCA learns a set of basis tokens with corresponding trajectories, which span diverse driving scenarios. Then, given any driving scene, it is able to probabilistically infer the future trajectory. By using RoCA together with a base E2E model in source-domain training, we improve the generalizability of the base model, without requiring extra inference computation. In addition, RoCA enables robust adaptation on new target domains, significantly outperforming direct finetuning. We extensively evaluate RoCA on various cross-domain scenarios and show that it achieves strong domain generalization and adaptation performance.

Deep Learning · Large Language Models

Xiaoyan Su, Peijie Dong, Zhenheng Tang, Song Tang, Yuyao Zhai, Kaitao Lin, Liang Chen, Gai Yuhang, Yuyu Luo, Qiang Wang 等

Despite the rapid advancements in Vision-Language Models (VLMs), a critical gap remains in their ability to handle structured, controllable diagrammatic tasks essential for professional workflows, as existing methods predominantly rely on pixel-based synthesis which operates in probabilistic pixel spaces and is inherently limited in editability and fidelity; instead, we propose a new "Diagram-as-Code" paradigm with symbolic logic that leverages `mxGraph` Extensible Markup Language (XML) for precise diagram generation and editing, and we present **VCG-Bench**, a unified benchmark for visual-centric `mxGraph` tasks comprising (1) a taxonomized dataset of 1,449 diverse diagrams spanning 6 domains and 15 sub-domains, (2) a paradigm definition that integrates Generation (Vision-to-Code) and Editability (Code-to-Code), and (3) a Tailored Evaluation Protocol employing multi-dimensional metrics such as `mxGraph` Execution Success Rate and Style Consistency Score (SCS), where experimental results highlight the challenges faced by current State-of-the-Art (SOTA) VLMs in structured fidelity and instruction compliance, reflecting their vision and reasoning capabilities.

Probabilistic Methods · Structure Learning

Shida Liu, Abhishek Gupta, Sumit Sinha, L Mahadevan

Symmetry is central to modern machine learning and physics: invariances and equivariances improve sample efficiency, robustness, and out-of-distribution generalization, while symmetry principles guide scientific modeling. Yet for stochastic dynamical systems, the relevant continuous symmetries are rarely known, and symmetry discovery for SDEs has remained essentially unexplored. We introduce *LieStoNet*, an end-to-end, *prior-free* framework for discovering Lie-point symmetries of SDEs directly from spatiotemporal trajectories, without prespecifying symmetry groups, templates, or canonical coordinates. Building on the seminal SDE Lie-symmetry theory of Gaeta and Quintero (1999), which formalizes Lie-point SDE symmetries and their relation to Fokker-Planck symmetries, LieStoNet learns neural surrogates for drift and diffusion from increments, then learns projectable generators by enforcing the SDE determining equations, separately regularizing for closure under Lie brackets, adherence to the Lie algebra axioms (bilinearity, antisymmetry, Jacobi), and a non-redundant independent basis. The surrogate also defines an associated Fokker-Planck equation, enabling optional discovery of its Lie-point symmetries in parallel. Across multiple canonical SDEs with known analytic symmetries, LieStoNet recovers generators consistent with the ground-truth symmetry algebra, providing interpretable symmetry discovery for noisy dynamics.

Probabilistic Methods · Everything Else

Ege Onur Taga, Samet Oymak, Shubhanshu Shekhar

We develop horizon-aware anytime-valid tests and confidence sequences for bounded means under a strict deadline $N$. Using the betting/e-process framework, we cast horizon-aware betting as a finite-horizon optimal control problem with state space $(t, \log W_t)$, where $t$ is the time and $W_t$ is the test martingale value. We first show that in certain interior regions of the state space, policies that deviate significantly from Kelly betting are provably suboptimal, while Kelly betting reaches the threshold with high probability. We then identify sufficient conditions showing that outside this region, more aggressive betting than Kelly can be better if log-wealth is low or time is short, and less aggressive can be better if log-wealth is high. Taken together these results suggest a simple phase diagram in the $(t, \log W_t)$ plane, delineating regions where Kelly, fractional Kelly, and aggressive betting may be preferable. Guided by this phase diagram, we introduce a Deep Reinforcement Learning approach based on a universal Deep Q-Network (DQN) agent that learns a single policy from synthetic experience and maps simple statistics of past observations to bets across horizons and null values. In limited-horizon experiments, the learned DQN policy outperforms state-of-the-art baselines.

Probabilistic Methods · Bayesian Models and Methods

Yujia Guo, Daolang Huang, Xinyu Zhang, Sammie Katt, Samuel Kaski, Ayush Bharti

Bayesian experimental design (BED) is a principled framework for data-efficient design of sequential experiments. However, existing BED methods are unable to adapt to dynamic constraints inherent in real-world tasks due to budget limitations, varying costs, or physical constraints that restrict how designs evolve over time. In this paper, we introduce a novel approach to BED that enables constrained optimization of experimental designs by combining offline pre-training of an amortized policy and a posterior network with online multi-step lookahead planning using scenario trees. We empirically demonstrate that our method yields substantially more informative design sequences than existing methods across a range of constrained BED tasks, while incurring only a modest additional computational overhead.

Theory · Probabilistic Methods

Junxi Xiao, Qinliang Su

Mean-field variational inference (VI) is scalable, but its independence assumption can severely limit inference when the posterior is inherently coupled across instances especially for correlated data. Existing structured VI approaches either impose simple dependence patterns or incur substantial cost as dependence becomes richer, leaving efficient higher-order instance-level dependence modeling largely unresolved. We propose Copula-SVI, which augments amortized marginals with an explicit vine-copula posterior and refines joint samples with Stein updates toward the true posterior. The vine construction makes dependence learning and sampling practical by decomposing it into bivariate copula factors, enabling edge-minibatched training with variance-aware level-wise sampling and efficient dependence-aware initialization via a sparse vine built from the same sampled edges. Experiments on constrained clustering and time series modeling show consistent improvements over strong structured VI baselines and demonstrate efficient higher-order instance-level dependence modeling.

Probabilistic Methods · Monte Carlo and Sampling Methods

Julia Linhart, Gabriel Cardoso, Alexandre Gramfort, Sylvain Le Corff, Pedro Luiz Coelho Rodrigues

Identifying the parameters of a non-linear model that best explain observed data is a core task across scientific fields. When such models rely on complex simulators, evaluating the likelihood is typically intractable, making traditional inference methods such as MCMC inapplicable. Simulation-based inference (SBI) addresses this by training deep generative models to approximate the posterior distribution over parameters using simulated data. In this work, we consider the tall data setting, where multiple independent observations provide additional information, allowing sharper posteriors and improved parameter identifiability. Building on the flourishing score-based diffusion literature, F-NPSE (Geffner et al., 2023) estimates the tall data posterior by composing individual scores from a neural network trained only for a single context observation. This enables more flexible and simulation-efficient inference than alternative approaches for tall datasets in SBI. However, it relies on costly Langevin dynamics during sampling. We propose a new algorithm that eliminates the need for Langevin steps by explicitly approximating the diffusion process of the tall data posterior. Our method retains the advantages of compositional score-based inference while being significantly faster and more stable than F-NPSE. We demonstrate its improved performance on toy problems and standard SBI benchmarks, and showcase its scalability by applying it to a complex real-world model from computational neuroscience.

Probabilistic Methods · Bayesian Models and Methods

Jinlin Lai, Antonio Linero, Yuling Yao

Vanilla variational inference finds an optimal approximation to the Bayesian posterior distribution, but even the exact Bayesian posterior is often not meaningful under model misspecification. We propose predictive variational inference (PVI): a general inference framework that seeks and samples from an optimal posterior density such that the resulting posterior predictive distribution is as close to the true data generating process as possible, while this closeness is measured by multiple scoring rules. By optimizing the objective, the predictive variational inference is generally not the same as, or even attempting to approximate, the Bayesian posterior, even asymptotically. Rather, we interpret it as implicit hierarchical expansion. Further, the learned posterior uncertainty detects heterogeneity of parameters among the population, enabling automatic model diagnosis. This framework applies to both likelihood-exact and likelihood-free models. We demonstrate its application in real data examples.

Probabilistic Methods · Bayesian Models and Methods

Qian Xie, Linda Cai, Alexander Terenin, Peter Frazier, Ziv Scully

In automated machine learning, scientific discovery, and other applications of Bayesian optimization, deciding when to stop evaluating expensive black-box functions in a cost-aware manner is an important but underexplored practical consideration. A natural performance metric for this purpose is the cost-adjusted simple regret, which explicitly captures the trade-off between solution quality and cumulative evaluation cost. Existing stopping rules for Bayesian optimization are either heuristic, or are theoretically grounded but designed to optimize simple regret without accounting for evaluation costs; as a result, they provide no guarantees against unnecessary evaluations when costs are high. We propose a principled cost-aware stopping rule for Bayesian optimization that adapts to varying evaluation costs without heuristic tuning. Our rule is grounded in a theoretical connection to state-of-the-art cost-aware acquisition functions, namely the Pandora's Box Gittins Index (PBGI) and log expected improvement per cost (LogEIPC). When paired with either acquisition function, we prove that the resulting policy satisfies a theoretical guarantee bounding the expected cost-adjusted simple regret. Across synthetic tasks and empirical benchmarks including hyperparameter optimization and neural architecture size search, pairing our stopping rule with PBGI or LogEIPC usually matches or outperforms other acquisition-function--stopping-rule pairs in terms of cost-adjusted simple regret.

Probabilistic Methods · Gaussian Processes

Jihao Andreas Lin, Sebastian Ament, Louis Tiao, David Eriksson, Maximilian Balandat, Eytan Bakshy

Gaussian processes (GPs) are powerful and widely used probabilistic regression models, but their effectiveness in practice is often limited by the choice of kernel function. This kernel function is typically handcrafted from a small set of standard functions, a process that requires expert knowledge, results in limited adaptivity to data, and imposes strong assumptions on the hypothesis space. We study Empirical GPs, a principled framework for constructing flexible, data-driven GP priors that overcome these limitations. Rather than relying on standard parametric kernels, we estimate the mean and covariance functions empirically from a corpus of historical observations, enabling the prior to reflect rich, non-trivial covariance structures present in the data. Theoretically, we show that the resulting model converges to the GP that is closest (in KL-divergence sense) to the real data generating process. Practically, we formulate the problem of learning the GP prior from independent datasets as likelihood estimation and derive an Expectation-Maximization algorithm with closed-form updates, allowing the model handle heterogeneous observation locations across datasets. We demonstrate that Empirical GPs achieve competitive performance on learning curve extrapolation and time series forecasting benchmarks.

Probabilistic Methods · Monte Carlo and Sampling Methods

Michael Van Huffel, Nathan Kirk, Makram Chahine, Daniela Rus, T. Konstantin Rusch

Low-discrepancy points are designed to efficiently fill the space in a uniform manner. This uniformity is highly advantageous in many problems in science and engineering, including in numerical integration, computer vision, machine perception, computer graphics, machine learning, and simulation. Whereas most previous low-discrepancy constructions rely on abstract algebra and number theory, Message-Passing Monte Carlo (MPMC) was recently introduced to exploit machine learning methods for generating point sets with lower discrepancy than previously possible. However, MPMC is limited to generating point sets and cannot be extended to low-discrepancy sequences (LDS), i.e., sequences of points in which every prefix has low discrepancy, a property essential for many applications. To address this limitation, we introduce Neural Low-Discrepancy Sequences *NeuroLDS*, the first machine learning-based framework for generating LDS. Drawing inspiration from classical LDS, we train a neural network to map indices to points such that the resulting sequences exhibit minimal discrepancy across all prefixes. To this end, we deploy a two-stage learning process: supervised approximation of classical constructions followed by unsupervised fine-tuning to minimize prefix discrepancies. We demonstrate that *NeuroLDS* outperforms all previous LDS constructions by a significant margin with respect to discrepancy measures. Moreover, we demonstrate the effectiveness of *NeuroLDS* across diverse applications, including numerical integration, robot motion planning, and scientific machine learning. These results highlight the promise and broad significance of Neural Low-Discrepancy Sequences.

Theory · Probabilistic Methods

Tao Wang, Edgar Dobriban

Prediction sets can wrap around any ML model to cover unknown test outcomes with a guaranteed probability. Yet, it remains unclear how to use them optimally for downstream decision-making. Here, we propose a decision-theoretic framework that seeks to minimize the expected loss (risk) against a worst-case distribution consistent with the prediction set's coverage guarantee. We first characterize the minimax optimal policy for a fixed prediction set, showing that it balances the worst-case loss inside the set with a penalty for potential losses outside the set. Building on this, we derive the optimal prediction set construction that minimizes the resulting robust risk subject to a coverage constraint. Finally, we introduce Risk-Optimal Conformal Prediction (ROCP), a practical algorithm that targets these risk-minimizing sets while maintaining finite-sample distribution-free marginal coverage. Empirical evaluations on medical diagnosis and safety-critical decision-making tasks demonstrate that ROCP reduces critical mistakes compared to baselines, particularly when out-of-set errors are costly.

Probabilistic Methods · Monte Carlo and Sampling Methods

Jennifer R. Andersson, Zheng Zhao

This paper is concerned with differentiable resampling in the context of sequential Monte Carlo (e.g., particle filtering). We propose a new informative resampling method that is instantly differentiable, based on an ensemble score diffusion model. We theoretically prove that our diffusion resampling method provides a consistent resampling distribution, and we show empirically that it outperforms the state-of-the-art differentiable resampling methods on multiple filtering and parameter estimation benchmarks. Finally, we show that it achieves competitive end-to-end performance when used in learning a complex dynamics-decoder model with high-dimensional image observations.