🤖 AI Summary
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.
📝 Abstract
MCMC algorithms are frequently used to perform inference under a Bayesian modeling framework. Convergence diagnostics, such as traceplots, the Gelman-Rubin potential scale reduction factor, and effective sample size, are used to visualize mixing and determine how long to run the sampler. However, these classic diagnostics can be ineffective when the sample space of the algorithm is highly discretized (eg. Bayesian Networks or Dirichlet Process Mixture Models) or the sampler uses frequent non-Euclidean moves. In this article, we develop novel generalized convergence diagnostics produced by mapping the original space to the real-line while respecting a relevant distance function and then evaluating the convergence diagnostics on the mapped values. Simulated examples are provided that demonstrate the success of this method in identifying failures to converge that are missed or unavailable by other methods.