Markov Chains for Exploring Posterior Distributions
by Luke Tierney
Technical Report No. 560 (Revised)
School of Statistics
University of Minnesota
March 15, 1994
Research supported in part by grant DMS-9005858 from the National Science
Foundation.
Abstract
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.