Discretization and MCMC Convergence Assessment
(Sprache: Englisch)
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...
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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.
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.
Inhaltsverzeichnis zu „Discretization and MCMC Convergence Assessment “
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.
Autoren-Porträt
Christian P. Robert is Professor of Statistics in the Applied Mathematics Department at the Université Paris Dauphine, and external lecturer at Ecole Polytechnique, Palaiseau, France. He was previously Head of the Statistics Laboratory at the Center for Research in Economics and Statistics (CREST) of the National Institute for Statistics and Economic Studies (INSEE) in Paris. In addition to many papers on Bayesian statistics, simulation methods, and decision theory, he has written three other books.
Bibliographische Angaben
- 1998, XI, 192 Seiten, Maße: 15,5 x 23,5 cm, Kartoniert (TB), Englisch
- Herausgegeben: Christian P. Robert
- Verlag: Springer, Berlin
- ISBN-10: 0387985913
- ISBN-13: 9780387985916
- Erscheinungsdatum: 13.08.1998
Sprache:
Englisch
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