Markov chains for exploring posterior distributions. (With discussion). (English) Zbl 0829.62080

Tierney, Luke

Ann. Stat. 22, No. 4, 1701-1762 (1994).

Summary: Several Markov chain methods are available for sampling from a posterior distribution. Two important examples are the Gibbs sampler and the Metropolis algorithm. In addition, several strategies are available for constructing hybrid algorithms. This paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues. On the theoretical side, results from the theory of general state space Markov chains can be used to obtain convergence rates, laws of large numbers and central limit theorems for estimates obtained from Markov chain methods. These theoretical results can be used to guide the construction of more efficient algorithms. For the practical use of Markov chain methods, standard simulation methodology provides several variance reduction techniques and also gives guidance on the choice of sample size and allocation.

MSC:

62M05	Markov processes: estimation; hidden Markov models
60J27	Continuous-time Markov processes on discrete state spaces
65C05	Monte Carlo methods
60J05	Discrete-time Markov processes on general state spaces

Keywords:

Monte Carlo; Metropolis-Hastings algorithm; Gibbs sampler; Metropolis algorithm; hybrid algorithms; general state space Markov chains; convergence rates; laws of large numbers; central limit theorems; simulation methodology; variance reduction; choice of sample size