mcmc diagnostics

In modern Bayesian inference, scalable MCMC methods (e.g., SGLD) introduce bias that invalidates conventional sample-quality diagnostics—such as effective sample size (ESS)—particularly for assessing multivariate dependence structures, a core inferential objective. To address this, we propose the Copula Discrepancy (CD) diagnostic, which leverages Sklar’s theorem to decouple and quantify fidelity of dependence structure in biased samples, establishing the first structural-aware framework for evaluating biased samplers. CD detects tail-dependence mismatches—even when Kendall’s tau agrees—thereby distinguishing divergent extremal event behaviors. We implement CD via moment estimation and a robust MLE variant, achieving significantly lower computational overhead than Stein-based alternatives. Experiments demonstrate that CD outperforms ESS and other standard metrics in hyperparameter selection: it precisely identifies optimal configurations and uncovers critical dependence biases missed by rank-correlation–based diagnostics.

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.

Bayesian model selection and averaging rely on the marginal likelihood, whose exact computation is intractable for complex models; standard estimators such as bridge sampling often yield high-variance approximations. This paper proposes a diagnostic, low-overhead framework to assess the reliability of marginal likelihood estimates: it introduces Pareto-$hat{k}$ diagnostics and block reordering into the bridge sampling pipeline, enabling robust quantification of Monte Carlo standard error (MCSE) without additional posterior sampling. The method integrates bridge sampling, MCSE estimation, and a dual-diagnostic mechanism. In simulation studies and real-world posterior distributions from posteriordb, it substantially reduces estimator variability and enhances credibility. The resulting tool provides a reproducible, verifiable, and practical solution for Bayesian model comparison.

This work addresses the slow MCMC convergence in Gibbs posterior sampling under stochastic loss functions, which stems from spurious dependence on the number of pseudo-observations. We propose the first pseudo-sample-size–independent corrected piecewise deterministic Markov process (PDMP) sampler. By designing a novel jump-rate function and direction mechanism, our method rigorously ensures that the invariant measure remains invariant to the pseudo-observation count—thereby overcoming the inherent trade-off between asymptotic bias and slow convergence in conventional stochastic-loss inference. We prove that the sampler converges exactly to the target Gibbs posterior measure with a uniform convergence rate independent of pseudo-sample size. Empirical validation across three canonical settings—likelihood-intractable models, misspecified models, and stochastic losses—demonstrates elimination of pseudo-sample-size bias in posterior sampling, alongside substantial improvements in robustness and estimation accuracy.

To address the unbounded time complexity and poor parallel scalability of conventional SMC/MCMC methods in Bayesian deep learning, this paper proposes a fully parallelized Sequential Monte Carlo (pSMC) framework. Our method achieves theoretically guaranteed strong scalability: mean squared error (MSE) scales as $O(1/NP)$, preserving constant per-step computational cost and zero efficiency loss as the number of processors $P o infty$. Key innovations include asynchronous distributed SMC, adaptive resampling, optimized inter-node sample communication, and rigorous convergence analysis. Experiments across multiple Bayesian inference tasks demonstrate that pSMC significantly outperforms state-of-the-art MCMC methods, attaining the optimal $O(varepsilon^{-2})$ computational complexity for $varepsilon$-accurate estimation, with empirically stable strong scaling behavior.

This work addresses the challenge of verifying ergodicity for adaptive MCMC algorithms in non-compact state and parameter spaces, where traditional compactness assumptions fail. By abandoning such restrictive assumptions, the authors instead introduce probabilistic bounds on the sample and parameter sequences to formulate a new set of easily verifiable sufficient conditions. Integrating tools from MCMC theory, adaptive algorithm analysis, and concentration inequalities, they establish a novel ergodicity framework that operates without compactness requirements. This approach significantly enhances the practical applicability and tractability of convergence analysis in complex real-world settings where non-compactness is inherent.

Approximate Bayesian inference often underestimates true uncertainty due to posterior credible intervals that are excessively narrow. This work proposes two simulation-based calibration (SBC)-driven methods for recalibrating approximate posteriors, systematically leveraging the SBC framework to adjust the width of posterior uncertainty intervals and achieve marginal calibration. The approach is applicable to complex model structures, including hierarchical models, and demonstrates consistent efficacy across diverse experimental settings by meaningfully widening posterior intervals. As a result, the proposed recalibration substantially enhances the calibration accuracy and reliability of approximate Bayesian inference.

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.

This work addresses the challenges in high-dimensional Bayesian regression, where conventional MCMC methods often get trapped in local modes and maximum a posteriori (MAP) estimation fails to quantify uncertainty. To overcome these limitations, the authors propose a hybrid approach that integrates deterministic optimization with stochastic sampling. Specifically, under a heavy-tailed hyperbolic error model, they first employ a two-stage ECM algorithm to efficiently perform variable selection and substantially reduce the model space. Subsequently, Gibbs sampling is conducted within the high posterior probability subspace to enable full posterior inference, complemented by Bayesian model averaging. The proposed method effectively balances computational efficiency, variable selection accuracy, and robust uncertainty quantification. Empirical evaluations on both simulated and real-world datasets demonstrate its superior performance over current state-of-the-art 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.