The exponential increase in the use of MCMC methods and the corre­ sponding applications in domains of even higher complexity have caused a growing concern about the available convergence assessment methods and the realization that some of these methods were not reliable enough for all-purpose analyses. Some researchers have mainly focussed on the con­ vergence to stationarity and the estimation of rates of convergence, in rela­ tion with the eigenvalues of the transition kernel. This monograph adopts a different perspective by developing (supposedly) practical devices to assess the mixing behaviour of the chain under study and, more particularly, it proposes methods based on finite (state space) Markov chains which are obtained either through a discretization of the original Markov chain or through a duality principle relating a continuous state space Markov chain to another finite Markov chain, as in missing data or latent variable models. The motivation for the choice of finite state spaces is that, although the resulting control is cruder, in the sense that it can often monitor con­ vergence for the discretized version alone, it is also much stricter than alternative methods, since the tools available for finite Markov chains are universal and the resulting transition matrix can be estimated more accu­ rately. Moreover, while some setups impose a fixed finite state space, other allow for possible refinements in the discretization level and for consecutive improvements in the convergence monitoring.

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.