Discretization and MCMC Convergence Assessment: (135 Lecture Notes in Statistics)

This monograph proposes several approaches to convergence monitoring for MCMC algorithms which are centered on the theme of discrete Markov chains. After a short introduction to MCMC methods, including recent developments like perfect simulation and Langevin Metropolis-Hastings algorithms, and to the current convergence diagnostics, the contributors present the theoretical basis for a study of MCMC convergence using discrete Markov chains and their specificities. The contributors stress in particular that this study applies in a wide generality, starting with latent variable models like mixtures, then extending the scope to chains with renewal properties, and concluding with a general Markov chain. They then relate the different connections with discrete or finite Markov chains with practical convergence diagnostics which are either graphical plots (allocation map, divergence graph, variance stabilizing, normality plot), stopping rules (normality, stationarity, stability tests), or confidence bounds (divergence, asymptotic variance, normality). Most of the quantitative tools take advantage of manageable versions of the CLT. The different methods proposed here are first evaluated on a set of benchmark examples and then studied on three full scale realistic applications, along with the standard convergence diagnostics: A hidden Markov modelling of DNA sequences, including a perfect simulation implementation, a latent stage modelling of the dynamics of HIV infection, and a modelling of hospitalization duration by exponential mixtures. The monograph is the outcome of a monthly research seminar held at CREST, Paris, since 1995. The seminar involved the contributors to this monograph and was led by Christian P. Robert, Head of the Satistics Laboratory at CREST and Professor of Statistics at the University of Rouen since 1992.

Table of Contents:
1 Markov Chain Monte Carlo Methods.- 1.1 Motivations.- 1.2 Metropolis-Hastings algorithms.- 1.3 The Gibbs sampler.- 1.4 Perfect sampling.- 1.5 Convergence results from a Duality Principle.- 2 Convergence Control of MCMC Algorithms.- 2.1 Introduction.- 2.2 Convergence assessments for single chains.- 2.3 Convergence assessments based on parallel chains.- 2.4 Coupling techniques.- 3 Linking Discrete and Continuous Chains.- 3.1 Introduction.- 3.2 Rao-Blackwellization.- 3.3 Riemann sum control variates.- 3.4 A mixture example.- 4 Valid Discretization via Renewal Theory.- 4.1 Introduction.- 4.2 Renewal theory and small sets.- 4.3 Discretization of a continuous Markov chain.- 4.4 Convergence assessment through the divergence criterion.- 4.5 Illustration for the benchmark examples.- 4.6 Renewal theory for variance estimation.- 5 Control by the Central Limit Theorem.- 5.1 Introduction.- 5.2 CLT and Renewal Theory.- 5.3 Two control methods with parallel chains.- 5.4 Extension to continuous state chains.- 5.5 Illustration for the benchmark examples.- 5.6 Testing normality on the latent variables.- 6 Convergence Assessment in Latent Variable Models: DNA Applications.- 6.1 Introduction.- 6.2 Hidden Markov model and associated Gibbs sampler.- 6.3 Analysis of thebIL67bacteriophage genome: first convergence diagnostics.- 6.4 Coupling from the past for theM1-M0model.- 6.5 Control by the Central Limit Theorem.- 7 Convergence Assessment in Latent Variable Models: Application to the Longitudinal Modelling of a Marker of HIV Progression.- 7.1 Introduction.- 7.2 Hierarchical Model.- 7.3 Analysis of the San Francisco Men’s Health Study.- 7.4 Convergence assessment.- 8 Estimation of Exponential Mixtures.- 8.1 Exponential mixtures.- 8.2 Convergence evaluation.- References.- Author Index.