coupling methods

This work addresses the challenge of diagnosing convergence in Markov chain Monte Carlo (MCMC) methods by proposing an efficient diagnostic framework based on multi-marginal coupling. By introducing shared randomness across multiple Metropolis–Hastings chains, the authors construct a Poisson Monte Carlo estimator and develop an adaptive point-process update rule alongside a distributed matching algorithm, substantially alleviating computational bottlenecks in high-dimensional settings. The approach establishes theoretical connections to list-level distribution coupling and distributed matching problems, yielding a natural optimization objective tailored for multi-chain coupling. Experimental results demonstrate that, across various dimensional configurations, the proposed method reduces coupling time by up to 50% compared to existing baselines, achieving significantly improved diagnostic efficiency.

There exists a long-standing disconnect between theoretical MCMC analysis—particularly via *f*-divergences—and practical convergence diagnostics: existing tools cannot directly monitor KL divergence, χ² divergence, Hellinger distance, or total variation distance. Method: We propose the first coupling-based *f*-divergence diagnostic framework, introducing a novel “weight coordination” mechanism that uniformly weights empirical measures from coupled chains, yielding a computable upper bound estimator for *f*-divergences that converges to zero as iterations increase. Our approach integrates coupled Markov chains, weighted empirical measures, and *f*-divergence theory, ensuring both theoretical guarantees and computational feasibility. Contribution/Results: Experiments demonstrate that our method accurately captures MCMC convergence dynamics and significantly outperforms state-of-the-art diagnostics on complex Bayesian inference tasks, providing a reliable, interpretable, and theoretically grounded standard for assessing MCMC convergence.

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.

Gaussian processes (GPs) struggle to rigorously incorporate uncountably infinite-dimensional functional prior information—such as boundary conditions or global physical constraints satisfied by PDE solutions. Method: This paper proposes a unified modeling framework grounded in reproducing kernel Hilbert spaces (RKHS), establishing for the first time a rigorous equivalence between the GP conditional expectation and orthogonal projection in RKHS. This enables direct embedding of functional constraints (e.g., Dirichlet or Neumann boundary conditions) into the GP prior, bypassing conventional pseudo-point approximations. Contribution/Results: We provide theoretical guarantees on existence, uniqueness, and convergence of the constrained GP posterior. Computationally, we design a practical numerical approximation algorithm. Experiments on PDE inverse problems demonstrate substantial improvements in uncertainty quantification accuracy and posterior consistency. The framework delivers a rigorous, general, and computationally tractable paradigm for integrating domain knowledge into Bayesian modeling.

This paper addresses the challenge of applying Yurinskii coupling under ℓₚ-norms (1 ≤ p ≤ ∞) to weakly approximable martingales. Methodologically, it introduces three innovations: (1) the first non-asymptotic ℓₚ-norm Yurinskii coupling for approximate martingales, substantially relaxing classical assumptions on independence, moment conditions, and exponential decay; (2) extension of coupling variables to generalized Gaussian mixtures, improving model flexibility; and (3) a novel third-order coupling scheme that sharply tightens approximation error bounds. The analysis integrates tools from mixing martingale theory, high-dimensional central limit theorems, empirical process theory, and local polynomial modeling. Key contributions include a central limit theorem for high-dimensional martingale vectors, a uniform strong Gaussian-mixture approximation for martingale empirical processes, and rigorous inferential guarantees—particularly for constructing confidence bands in nonparametric regression.

This work establishes novel maximal inequalities for empirical processes under graph-dependent observations, revealing that their convergence rates are jointly governed by the complexity of the function class, the underlying graph structure, and the decay of dependence. By integrating graph coloring, block partitioning, and graph-adapted dependence coefficients, the authors develop a coupling strategy that effectively disentangles the complexity of the indexing class from the graph-induced dependence, thereby overcoming the classical √n convergence barrier. The theoretical framework accommodates graphs with polynomial or exponential growth as well as directed binary networks, yielding Glivenko–Cantelli-type results and explicit quantifications of effective sample size. These advances enable uniform laws of large numbers in applications such as network autoregressive models, nonlinear local propagation dynamics, and settings involving treatment interference.

This study systematically traces the century-long evolution of the Fréchet distance—from its original 1906 formulation in abstract metric spaces and its 1957 reinterpretation via couplings of probability measures to its 1990s adaptation as a curve-matching algorithm—thereby establishing, for the first time, a unified theoretical bridge between its geometric and probabilistic interpretations. Through historical document analysis and optimal transport theory, the work demonstrates that the Fréchet Inception Distance (FID), widely used to evaluate generative models, is in fact a special case of the Wasserstein-2 distance operating in the feature space induced by deep neural networks. Beyond clarifying the historical and conceptual trajectory of the Fréchet distance, this research provides an English translation appendix of key original texts and offers a deeper theoretical grounding for understanding FID as a metric in modern machine learning.

This work addresses the challenge of quantifying spatial clustering of probability distributions over graph-structured data by introducing a diffusion distance that incorporates global graph geometry. Built upon a Metropolis–Hastings Markov chain, the proposed measure characterizes the rate at which a distribution converges to uniformity to assess clustering intensity, thereby extending the classical Moran’s I statistic to a global perspective. Theoretically, the distance is linked to spectral graph theory and optimal transport, with formal stability guarantees provided. Empirical evaluations demonstrate superior statistical power over Moran’s I in synthetic benchmarks and reveal nuanced segregation patterns in the distribution of Black populations across 100 U.S. cities—patterns overlooked by traditional local indicators.

This work addresses the computational challenges of evaluating Wasserstein distances in large-scale non-Euclidean spaces by proposing a projection-based approach grounded in 1-Lipschitz observables. The method pushes forward probability measures onto the real line, computes their one-dimensional Wasserstein distances, and constructs a hierarchy of pseudometrics over nested subspaces to approximate the original distance. This hierarchical framework balances accuracy and efficiency through tunable parameters and establishes a theoretical link between the metric covering dimension of the support set and the order required for unique measure recovery—providing an analogue of the Cramér–Wold theorem in non-Euclidean settings. Theoretical analysis confirms that measures can be uniquely recovered at specific hierarchy levels, and numerical experiments on finite discrete grids demonstrate the method’s effectiveness and practicality.

This work proposes a unified mathematical framework grounded in dynamic information flow for constructing structurally rigorous models of future prediction. By integrating filtering theory, regular conditional probabilities, Markov semigroups, infinitesimal generators, and multiple information geometries—including Hilbert, Fisher–Rao, and Wasserstein—the approach conceptualizes prediction as the construction of conditional distributions governed by informational, geometric, and modeling constraints. The framework elucidates deep connections among classical results such as the tower property and semigroup laws, as well as Itô’s formula and backward equations. Explicit transition laws, spectral decompositions, term structures, and asymptotic behaviors are derived within canonical models like Ornstein–Uhlenbeck and Cox–Ingersoll–Ross, thereby establishing a compact mathematical mapping from idealized theoretical constructs to empirical forecasting.