Discretization and MCMC convergence assessment

This monograph explores convergence monitoring for MCMC algorithms, focusing on discrete Markov chains. It begins with an overview of MCMC methods, including recent advancements like perfect simulation and Langevin Metropolis-Hastings algorithms, as well as current convergence diagnostics. The contributors establish a theoretical framework for studying MCMC convergence through discrete Markov chains, emphasizing its broad applicability, from latent variable models such as mixtures to chains with renewal properties and general Markov chains. They connect these concepts with practical convergence diagnostics, which include graphical plots (allocation maps, divergence graphs, variance stabilizing plots, normality plots), stopping rules (normality, stationarity, stability tests), and confidence bounds (divergence, asymptotic variance, normality). Many quantitative tools leverage manageable versions of the Central Limit Theorem (CLT). The proposed methods are evaluated using benchmark examples and three realistic applications: hidden Markov modeling of DNA sequences with perfect simulation, latent stage modeling of HIV infection dynamics, and modeling hospitalization duration through exponential mixtures. This work stems from a monthly research seminar at CREST, Paris, initiated in 1995, led by Christian P. Robert, a prominent figure in the field.