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Markov chain Monte Carlo methods. (Méthodes de Monte Carlo par chaînes des Markov.) (French) Zbl 0917.60007
Statistique mathématique et probabilité. Paris: Éditions Économica. x, 340 p. (1996).
This book exposes the Markov chain Monte Carlo (MCMC) method richness in the general context of the simulation theory and the particular aspects of its application to statistics. The contents of Christian Robert’s work provides the theoretical basis of the MCMC method in a rigorous manner, as well as the state-of-the-art of MCMC application to the various subdomains of statistics. The examples comprised illustrate particularly the MCMC method in Bayesian parametric inference, but there are other different fields in which MCMC may be successfully applied: Bayesian treatment of images, non-parametric estimation of curves, calibration of neural networks, maximum likelihood estimation on censored data, etc.
The seven chapters of the book are organized as follows: Chapter 1 (Usual methods in simulation) exposes the theoretical bases of simulation methods. Chapter 2 (Monte Carlo method and stochastic optimization) presents the application of simulation to integration and stochastic optimization, including comparisons between Monte Carlo methods (e.g. weighted sampling). Chapter 3 (Markov chains: Stability and convergence) is devoted to the theoretical aspects of Markov chains. Chapter 4 (MCMC methods: Hastings-Metropolis algorithms) and Chapter 5 (MCMC methods: Gibbs’ sampling) comprise two major types of MCMC algorithms together with optimization, control and convergence of these algorithms. The same topics receive a special attention in Chapter 6 (Convergence control of MCMC algorithms). Chapter 7 (Models of missing data) provides useful application examples of the theories in the previous chapters to various models of missing data. A very rich bibliography and useful pointers to topics and authors are included.
Reviewer: N.Curteanu (Iaşi)

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
65C05 Monte Carlo methods
68U20 Simulation (MSC2010)