Discretization and MCMC Convergence Assessment / Edition 1

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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.

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Product Details

  • ISBN-13: 9780387985916
  • Publisher: Springer New York
  • Publication date: 8/13/1998
  • Series: Lecture Notes in Statistics Series, #135
  • Edition description: Softcover reprint of the original 1st ed. 1998
  • Edition number: 1
  • Pages: 192
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.44 (d)

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.- 2.2.1 Graphical evaluations.- 2.2.2 Binary approximation.- 2.3 Convergence assessments based on parallel chains.- 2.3.1 Introduction.- 2.3.2 Between-within variance criterion.- 2.3.3 Distance to the stationary distribution.- 2.4 Coupling techniques.- 2.4.1 Coupling theory.- 2.4.2 Coupling diagnoses.- 3 Linking Discrete and Continuous Chains.- 3.1 Introduction.- 3.2 Rao-Blackwellization.- 3.3 Riemann sum control variates.- 3.3.1 Merging numerical and Monte Carlo methods.- 3.3.2 Rao-Blackwellized Riemann sums.- 3.3.3 Control variates.- 3.4 A mixture example.- 4 Valid Discretization via Renewal Theory.- 4.1 Introduction.- 4.2 Renewal theory and small sets.- 4.2.1 Definitions.- 4.2.2 Renewal for Metropolis-Hastings algorithms.- 4.2.3 Splitting the kernel.- 4.2.4 Splitting in practice.- 4.3 Discretization of a continuous Markov chain.- 4.4 Convergence assessment through the divergence criterion.- 4.4.1 The divergence criterion.- 4.4.2 A finite example.- 4.4.3 Stopping rules.- 4.4.4 Extension to continuous state spaces.- 4.4.5 From divergence estimation to convergence control.- 4.5 Illustration for the benchmark examples.- 4.5.1 Pump Benchmark.- 4.5.2 Cauchy Benchmark.- 4.5.3 Multinomial Benchmark.- 4.5.4 Comments.- 4.6 Renewal theory for variance estimation.- 4.6.1 Estimation of the asymptotic variance.- 4.6.2 Illustration for the Cauchy Benchmark.- 4.6.3 Finite Markov chains.- 5 Control by the Central Limit Theorem.- 5.1 Introduction.- 5.2 CLT and Renewal Theory.- 5.2.1 Renewal times.- 5.2.2 CLT for finite Markov chains.- 5.2.3 More general CLTs.- 5.3 Two control methods with parallel chains.- 5.3.1 CLT and Berry-Esséen bounds for finite chains.- 5.3.2 Convergence assessment by normality monitoring.- 5.3.3 Convergence assessment by variance comparison.- 5.3.4 A finite example.- 5.4 Extension to continuous state chains.- 5.4.1 Automated normality control for continuous chains.- 5.4.2 Variance comparison.- 5.4.3 A continuous state space example.- 5.5 Illustration for the benchmark examples.- 5.5.1 Cauchy Benchmark.- 5.5.2 Multinomial Benchmark.- 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.- 6.2.1M1-M0hidden Markov model.- 6.2.2 MCMC implementation.- 6.3 Analysis of thebIL67bacteriophage genome: first convergence diagnostics.- 6.3.1 Estimation results.- 6.3.2 Assessing convergence with CODA.- 6.4 Coupling from the past for theM1-M0model.- 6.4.1 The CFTP method.- 6.4.2 The monotone CFTP method for theM1-M0DNA model.- 6.4.3 Application to thebIL67bacteriophage.- 6.5 Control by the Central Limit Theorem.- 6.5.1 Normality control for the parameters with parallel chains.- 6.5.2 Testing normality of the hidden state chain.- 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.- 7.2.1 Longitudinal disease process.- 7.2.2 Model of marker variability.- 7.2.3 Implementation.- 7.3 Analysis of the San Francisco Men’s Health Study.- 7.3.1 Data description.- 7.3.2 Results.- 7.4 Convergence assessment.- 8 Estimation of Exponential Mixtures.- 8.1 Exponential mixtures.- 8.1.1 Motivations.- 8.1.2 Reparameterization of an exponential mixture.- 8.1.3 Unknown number of components.- 8.1.4 MCMC implementation.- 8.2 Convergence evaluation.- 8.2.1 Graphical assessment.- 8.2.2 Normality check.- 8.2.3 Riemann control variates.- References.- Author Index.

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