probability theory

Discrete probabilistic sampling algorithms lack formal verification frameworks. Method: This paper proposes the first verification methodology based on *distributional loop invariants*: probabilistic programs are modeled as distribution transformers, and a Hoare-style logic is developed to support both total and partial correctness proofs. Contribution/Results: The key innovation is the systematic introduction of *distributional loop invariants*, enabling precise characterization and inductive reasoning about the evolution of output distributions during program execution. The framework unifies probabilistic program semantics, invariant inference, and formal verification techniques. Experimentally, it successfully verifies the correctness of two classic algorithms—Fast Dice Roller and Fast Loaded Dice Roller—demonstrating strong expressive power and practical applicability. This work establishes a novel paradigm for formal verification of probabilistic programs.

This work addresses the quantitative verification of probabilistic programs and stochastic dynamical systems, specifically aiming to rigorously infer upper bounds on the probability that a stochastic process reaches a target condition within a finite number of steps. We propose a neuro-symbolic approach: supermartingale certificates are parameterized using differentiable neural networks; training employs stochastic optimization, while formal verification leverages SMT solvers (e.g., Z3); and an counterexample-guided inductive synthesis (CEGIS) framework enables iterative refinement. To our knowledge, this is the first method to embed neural networks directly into supermartingale construction—balancing expressive power with formal verifiability—and thereby significantly improves bound tightness and reliability. Evaluated on diverse benchmarks, our computed probability bounds match or surpass those of state-of-the-art techniques. Notably, we successfully verify high-dimensional, nonlinear stochastic models that defy analysis by conventional symbolic methods.

Statistical Model Checking (SMC) often yields inflated error rates in probabilistic and expected reward estimation due to insufficient statistical rigor. To address this, we propose a robust estimation framework with rigorous theoretical guarantees: (i) we extend the Dvoretzky–Kiefer–Wolfowitz (DKW) inequality to expected reward estimation for the first time; (ii) we introduce a limit-PAC (Probably Approximately Correct) procedure ensuring controllable estimation error; and (iii) we derive a computable upper bound on reachability rewards and enhance practicality via path truncation and distribution bounding. Our method is implemented in the *modes* tool. Experimental evaluation demonstrates a substantial reduction in erroneous conclusions while maintaining high precision, thereby ensuring both statistical correctness and engineering applicability.

For Markov decision processes (MDPs) with unknown transition probabilities, existing statistical model checking (SMC) algorithms suffer from high sample complexity and weak theoretical guarantees. Method: We introduce tight concentration inequalities—specifically, the Bretagnolle–Huber and Empirical Bernstein bounds—into the SMC framework for the first time, and design adaptive, structure-aware statistical estimators that exploit MDP topology. Contribution/Results: Theoretically, our approach yields significantly tighter and more general probably approximately correct (PAC) guarantees. Empirically, it reduces required sample sizes by up to two orders of magnitude on standard verification benchmarks. This work establishes a new paradigm for efficient and reliable formal verification of uncertain systems.

This paper addresses the challenge of modeling epistemic uncertainty in hypothesis testing under partial ignorance. We propose the first nonparametric two-sample testing framework tailored to credal sets—convex sets of probability measures. Methodologically, we introduce a novel permutation-testing paradigm specifically designed for finitely generated credal sets, directly incorporating epistemic uncertainty into the testing procedure; further, we develop a nuisance-parameter correction mechanism and integrate kernel methods with convex probability measure theory. Our contributions are threefold: (i) the first framework supporting structured hypothesis tests—including equivalence, inclusion, intersection, and disjointness—over credal sets; (ii) significantly enhanced robustness and reliability under model uncertainty; and (iii) empirical validation on real-world data, with natural extensibility to multi-distribution credal sample comparison.

This work aims to elucidate the empirical success and performance origins of intelligent theorem-proving systems on classical proof search problems. To this end, we model the proving process as a time-constrained Markov decision process and introduce the notion of “statistical provability,” which characterizes—through a distributional lens—the probability that a system generates a valid proof within a bounded number of steps. Leveraging the Bellman equation, we establish the existence of an optimal policy and, for the first time, quantify the performance gap of score-guided planning methods. Through analyses involving sub- and super-solution inequalities, metric entropy, doubling structures, and tail bounds on action gaps, our study not only provides a theoretical justification for the effectiveness of intelligent provers under realistic, biased distributions but also reveals their fundamental limitations in worst-case or adversarial scenarios.

This work addresses the inefficiency of existing sampling-based inference methods, which struggle to effectively explore critical decision points during resampling. The authors propose a training-free inference optimization approach that identifies significant entropy spikes in the base model’s next-token prediction along the reasoning trajectory as proxies for key decision points. By integrating an entropy-guided cutoff selection strategy with the Metropolis-Hastings sampling algorithm, the method enables targeted resampling that substantially reduces mixing time complexity. Evaluated on multiple challenging benchmarks—including MATH500, HumanEval, GPQA Diamond, and AIME26—the approach consistently outperforms current baselines and even reinforcement learning–trained models, demonstrating strong training-free reasoning capabilities.

Distribution verification of samplers over infinite domains (e.g., the natural numbers) has long been hindered by worst-case computational complexity, rendering existing methods impractical. This paper introduces the first instance-dependent, efficient testing framework, built upon an interval-conditioning mechanism that jointly estimates continuous distribution mass and the distance between unknown and known distributions—thereby decoupling test efficiency from worst-case guarantees. The method supports rigorous statistical verification of samplers over arbitrary countably infinite domains while ensuring computational tractability. Empirical evaluation demonstrates up to 1000× speedup over state-of-the-art approaches across diverse distribution families, significantly improving practicality, scalability, and deployability of sampler validation.

This work addresses the challenge of efficiently sampling from complex probability distributions in generative AI by formalizing “test-time training” (TTT) as the problem of generating samples from a restricted class of distributions, given an oracle that provides approximate density evaluations of the target distribution. By integrating the classical reduction from approximate counting to sampling, query complexity analysis, and an online adaptation mechanism with feedback, the paper establishes the first theoretical framework for TTT. Its core contributions include proving the optimality of the Jerrum–Sinclair random walk in the general setting and demonstrating that, when the distribution class is suitably restricted, one can surpass the quadratic query complexity lower bound inherent to generic sampling methods, thereby enabling more efficient sampling strategies.

Can base language models (LLMs) achieve reasoning capabilities comparable to reinforcement learning (RL)-fine-tuned models—without any additional training, solely through inference-time sampling? Method: We propose an MCMC-inspired iterative sampling algorithm that constructs a Markov chain using the base model’s own token-level likelihoods, dynamically concentrating sampling effort on high-probability reasoning paths during inference. Contribution/Results: This work provides the first empirical evidence that strong latent reasoning capacity exists intrinsically in base LLMs and can be unlocked purely via sampling. Unlike RL-based methods, our approach avoids diversity collapse, requires no labeled data, reward models, validators, or parameter updates. On single-sample reasoning benchmarks—including MATH500, HumanEval, and GPQA—it matches or exceeds the performance of RLHF- and GRPO-fine-tuned models, while preserving rich sample diversity across multiple generations. The method is fully general, training-free, and deployment-friendly.