partition function analysis

This paper presents the first efficient distributed approximation algorithm for computing the partition function of graphical Gibbs distributions in the Congested Clique model, unifying counting problems for combinatorial structures such as $q$-colorings and independent sets. Methodologically, it introduces, for the first time in this model, the theoretical connection between combinatorial sampling and counting, establishing a generic framework grounded in locality and rapid mixing conditions; it further enables parallel Markov chain sampling via triangle counting and semiring matrix multiplication. Theoretical contributions include: (i) an $widetilde{O}(n^{1/3}/varepsilon^2)$-round algorithm for $q$-coloring counting when $q > 2Delta$; and (ii) convergence to $varepsilon$-accuracy for the hard-core model within $widetilde{O}(1/varepsilon^2)$ rounds under fugacity $lambda leq alpha/(Delta-1)$ with $alpha < 1$, substantially improving upon prior distributed results.

This work addresses the detection of a hidden matching—a set of disjoint planted edges—in an Erdős–Rényi random graph, formulated as a binary hypothesis testing problem, and precisely characterizes its statistical detectability threshold. Methodologically, it introduces the cluster expansion technique from statistical physics to perform non-asymptotic, exact analysis of the log-likelihood ratio—a first for planted models—and integrates this with fine-grained analysis of monomer-dimer partition function differences, combinatorial cancellation arguments, and edge/wedge counting statistics. The analysis rigorously establishes that no statistical-computational gap exists for this problem. As a result, the optimal signal-to-noise ratio threshold is identified, and a polynomial-time detector achieving the information-theoretic limit is explicitly constructed. This unifies statistical feasibility and computational tractability, closing the gap between what is statistically possible and what is efficiently computable.

This study addresses the lack of efficient sampling and partition function approximation algorithms for ferromagnetic two-state systems within certain parameter regimes. By constructing weighted subgraph and random-cluster-like models, the authors establish novel equivalences between these combinatorial structures and the target physical system. Leveraging these connections, they propose the first efficient sampling algorithm tailored to this regime and design a partition function approximation algorithm running in nearly quadratic time on bounded-degree graphs and in polynomial time on general graphs. This represents a significant improvement over the method of Guo et al. (2020). The work integrates techniques from graph theory, statistical physics, and randomized approximation to deliver an enhanced computational framework for analyzing such systems.

This paper establishes an axiomatic unification framework bridging information theory and statistical thermodynamics. Method: It introduces a novel synthesis of large deviations theory, Kolmogorov conditional expectation, and information projection, using empirical frequencies as the foundational driver to construct an extended information geometry; entropy functions—including Shannon entropy, mutual information, and relative entropy—are systematically derived and endowed with thermodynamic energy interpretations via Legendre–Fenchel parametrization of the empirical mean manifold. Contribution/Results: (1) It reveals thermodynamic-style additivity and intrinsic Riemannian geometric structure of entropies in the infinite-sample limit; (2) it unifies the additive properties of the empirical mean manifold with those of statistical thermodynamics; (3) it fundamentally extends information geometry from the space of probability distributions to the space of empirical frequencies, thereby providing a new paradigm for the geometric and physical interpretation of information.

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.

This work investigates the efficient characterization of the evolution of input probability density distributions in neural networks within the framework of probabilistic abstract interpretation. To overcome the representational limitations of traditional grid-based abstractions, the authors introduce two novel abstract domains—distributional approximation and clustering approximation—and design corresponding abstract transformers to formally analyze neural network behavior. This approach represents the first integration of these approximation mechanisms into probabilistic abstract interpretation, offering theoretical soundness guarantees. Empirical evaluations demonstrate its superior precision and expressive power compared to existing methods, thereby establishing a new pathway for the formal verification of neural networks.

This study addresses the computational challenges in Bayesian inference for statistical models involving intractable normalizing functions. It provides a systematic review and comparison of mainstream inference algorithms, including Markov chain Monte Carlo (MCMC), approximate Bayesian computation, pseudo-likelihood, and general likelihood-free methods. The work innovatively constructs a diagnostic framework to assess the accuracy of approximate algorithms, thereby enhancing the reliability of algorithmic tuning. By elucidating the intrinsic connections and applicability boundaries among these approaches, the paper clarifies the trade-offs between theoretical properties and empirical performance. Furthermore, it offers practitioners a clear guideline for algorithm selection, significantly improving the feasibility and credibility of inference in complex models with intractable normalizing constants.

This study addresses the challenges of statistical inference for the Ising model on inhomogeneous random graphs in the subcritical regime, where maximum likelihood estimation is computationally infeasible and asymptotic optimality theory is lacking. The authors develop a unified statistical inference framework applicable to both sparse and dense networks, establishing—for the first time—the local asymptotic minimax theorem and its associated information-theoretic limits for network-dependent data. They propose a computationally efficient one-step closed-form estimator whose asymptotic distribution coincides with that of the maximum likelihood estimator. This estimator achieves the optimal asymptotic risk rate and constant, enabling the construction of valid confidence intervals and yielding sharp characterizations of the local power and minimax detection rate for goodness-of-fit tests of natural parameters.

This work addresses the problem of efficiently estimating the partition function given only samples from a proposal distribution and access to the unnormalized density ratio of the target distribution. To this end, it introduces the novel concept of an “integral coverage profile,” which characterizes the concentration of the target distribution’s mass in regions of high density ratio. A general information-theoretic framework is established, relying solely on the f-divergence between the proposal and target distributions, and applicable to broad settings including heavy-tailed distributions. Under minimal assumptions, the framework rigorously distinguishes the complexity of approximate sampling from that of counting. By combining a generalized Paley–Zygmund inequality with importance sampling analysis, the paper derives tight upper and lower bounds on the sample complexity for multiplicative estimation of the partition function, unifying and extending classical results on importance sampling, rejection sampling, and heavy-tailed mean estimation, while providing sharper finite-sample guarantees.

This work addresses the challenge of integrating logical interpretability with probabilistic reasoning in high-stakes decision-making. It proposes a novel probabilistic state algebra that uniquely embeds logical reduction directly within purely linear algebraic operations, eliminating the need for graph traversal or circuit compilation. By mapping logical states to energy potentials and employing the Hadamard product to construct the Gibbs distribution of a Markov random field, the framework unifies symbolic rules with statistical inference. The approach supports modular rule representation using t-objects and wildcards, yielding a mathematically rigorous, auditable, and maintainable probabilistic logic system well-suited for high-risk human-AI collaboration domains such as healthcare and finance.