perform mcmc diagnostics

Existing MCMC convergence diagnostics—such as trace plots, the Gelman–Rubin statistic, and effective sample size (ESS)—frequently fail or are ill-defined in discrete or non-Euclidean parameter spaces (e.g., Bayesian networks, Dirichlet process mixture models). To address this limitation, we propose a novel, generalized diagnostic paradigm: first, apply distance-preserving embedding to map the original space into the real line; then, reconstruct trace plots and adapt the Gelman–Rubin statistic and ESS estimation in the embedded space. This constitutes the first systematic framework for MCMC convergence assessment specifically designed for discrete and non-Euclidean parameter spaces, thereby overcoming the traditional reliance on continuous Euclidean geometry. Extensive simulation studies demonstrate that our method robustly detects convergence failures missed by standard diagnostics, significantly enhancing the reliability and diagnosability of MCMC inference in non-Euclidean models.

Introductory teaching resources for Markov chain Monte Carlo (MCMC) methods—particularly Gibbs sampling—lack beginner-friendly, hands-on tools. Method: This paper introduces a reproducible pedagogical framework implemented in R, tightly integrating theoretical exposition with line-by-line executable code. Designed progressively, it covers foundational topics including probabilistic modeling, transformation of random variables, numerical integration, and Gibbs sampler implementation. Contribution/Results: Diverging from conventional lecture notes or opaque software packages, the framework pioneers a “theory–code–visualization” triadic pedagogy, supported by comprehensive case studies and an interactive learning environment. Empirical evaluation demonstrates significant improvements in beginners’ conceptual understanding and practical proficiency regarding sampling mechanisms, convergence diagnostics, and posterior inference. The framework fills a critical gap by providing a lightweight, highly accessible, and pedagogically grounded MCMC teaching tool.

This study addresses the limitation of standard MCMC diagnostics, which frequently misinterpret multimodal posteriors as sampling failures. To overcome this, it proposes a graph-theoretic convergence diagnostic framework that constructs a graph structure by computing between-chain R-hat values, thereby identifying groups of Markov chains exploring the same posterior mode. The method is implemented using Stan and the R package mcmcConvergenceGraph. As the first approach to leverage graph structures for distinguishing genuine multimodality from poor mixing, it prevents the penalizing misdiagnosis of multimodal posteriors. By providing both graphical and numerical summaries, the framework offers comprehensive diagnostic insights. Its effectiveness in accurately diagnosing multimodal behavior is validated through applications in regression and pharmacokinetic models.

In Bayesian sequential trials, error rate evaluation relies on computationally expensive Monte Carlo simulations, hindering efficient optimization of sample size and decision thresholds. Method: This paper establishes, for the first time, analytical functional relationships between posterior and posterior predictive probabilities and sample size. Leveraging Bayesian decision theory and asymptotic analysis—combined with numerical fitting and error-rate inversion—the method enables precise error-rate assessment for any sample size using only two simulations, and rapidly identifies optimal design parameters. Contribution/Results: The approach drastically reduces computational cost while achieving error-rate control accuracy comparable to conventional simulation-based methods. In two real-world case studies, it attains exact error-rate calibration and accelerates design optimization by several orders of magnitude. This provides a scalable, verifiable, and highly efficient design paradigm for Bayesian adaptive trials.

This work addresses the challenge of assessing convergence rates of Markov chain Monte Carlo (MCMC) algorithms. We propose a novel, computable upper bound on the Wasserstein distance between the current and stationary distributions, leveraging Common Random Numbers (CRN) in simulation. Our method is the first to systematically integrate CRN into MCMC convergence diagnostics: by coupling two chains driven by identical random inputs, it yields a tight, analytically tractable upper bound on the convergence rate. Theoretical analysis and empirical evaluation—across canonical Bayesian settings including variance-components models and James–Stein correlation models—demonstrate that the CRN-based bound substantially outperforms classical drift/minorization bounds in both tightness and practicality. Crucially, our bound is both theoretically grounded and computationally feasible, providing the first quantitative convergence criterion for MCMC practitioners. This advances MCMC diagnostics from qualitative heuristics toward rigorous, quantitative assessment.

This study addresses the limitation of traditional statistical process control, which emphasizes detecting historical shifts while neglecting the acceptability of the current process state. To overcome this, the authors propose a Bayesian sequential monitoring framework tailored for recoverable processes subject to parameter drift. By recursively computing the posterior probability that the process is in-control at the current time, the method shifts the monitoring focus toward real-time state assessment. The framework integrates time-to-failure modeling, Gaussian and binomial tracking, and multivariate data analysis within a unified Bayesian formulation. Demonstrated through simulation studies and an application to white wine quality data, the approach effectively identifies the current operational status of dynamic recoverable processes, significantly enhancing both monitoring accuracy and practical applicability.

本文探讨了贝叶斯样本量确定的两种方法：估计抽样分布或通过随机根查找探索，评估了它们在复杂模型中的性能。

This work addresses the inefficiency and lack of a unified theoretical understanding of gradient-based MCMC methods when sampling from anisotropic and hierarchical posteriors. Starting from continuous-time dynamics, the authors systematically derive algorithms such as HMC, MALA, NUTS, and MAKLA through numerical discretization and Metropolis correction, establishing a cohesive theoretical framework. They introduce two key innovations: a globally whitened mass matrix and a stochastic step-size strategy, which together mitigate sampling difficulties arising from state-dependent curvature. The proposed approach substantially improves sampling efficiency and demonstrates superior convergence and stability in large-scale Bayesian inference and hierarchical models.

This work addresses the challenge of posterior inference in chained Markov fusion models, where shared variables between adjacent submodels complicate computation. To overcome this, the authors propose a multi-stage sequential Monte Carlo (SMC) sampler based on a divide-and-conquer strategy. This approach introduces divide-and-conquer SMC into the chained Markov fusion framework for the first time, decomposing the model using a tree structure to enable flexible composition of an arbitrary number of heterogeneous submodels and efficient Bayesian inference. Experimental results demonstrate that the method accurately estimates key parameters—such as immigration and reproduction rates—in both a synthetic example comprising 11 submodels and an ecological integrated population model, significantly improving inference efficiency and scalability.

Traditional hybrid experimental designs struggle to robustly control the frequentist operating characteristics of Bayesian decisions under model misspecification and lack efficient sample size determination methods applicable to generalized posteriors. This work proposes a computationally efficient experimental design framework that requires simulations at only two sample sizes and leverages extrapolation modeling of posterior summary functions to infer performance across the entire sample size space. This approach enables identification of the minimal sample size and decision rule satisfying desired operating characteristics. It represents the first general and scalable method for sample size planning under generalized posteriors, substantially reducing computational burden while enhancing robustness to model misspecification. The method’s validity and broad applicability within Bayesian M-estimation–type experiments are demonstrated through the redesign of an adaptive clinical trial with time-to-event outcomes.