Discretization and MCMC Convergence Assessment

1. Front Matter

   Pages i-xi

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2. Markov Chain Monte Carlo Methods

   • Christian P. Robert, Sylvia Richardson

   Pages 1-25

3. Convergence Control of MCMC Algorithms

   • Christian P. Robert, Dominique Cellier

   Pages 27-46

4. Linking Discrete and Continuous Chains

   • Anne Philippe, Christian P. Robert

   Pages 47-66

5. Valid Discretization via Renewal Theory

   • Chantal Guihenneuc-Jouyaux, Christian P. Robert

   Pages 67-97

6. Control by the Central Limit Theorem

   • Didier Chauveau, Jean Diebolt, Christian P. Robert

   Pages 99-126

7. Convergence Assessment in Latent Variable Models: DNA Applications

   • Florence Muri, Didier Chauveau, Dominique Cellier

   Pages 127-146

8. Convergence Assessment in Latent Variable Models: Application to the Longitudinal Modelling of a Marker of HIV Progression

   • Chantal Guihenneuc-Jouyaux, Sylvia Richardson, Virginie Lasserre

   Pages 147-160

9. Estimation of Exponential Mixtures

   • Marie-Anne Gruet, Anne Philippe, Christian P. Robert

   Pages 161-173

10. Back Matter

    Pages 175-194

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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.

• Latent variable model
• Markov chain
• Markov model
• Variance
• algorithms
• renewal theory
• statistics

• INSEE Crest, Malakoff Cedex, France

  Christian P. Robert

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