Markov Chain Monte Carlo Estimation of the Law of the Mean of a Dirichlet Process
Alessandra Guglielmi, Richard L. Tweedie · Bernoulli · 2001
The distribution \\cal{M}}α of the mean Γα of a Dirichlet process on the real line, with parameter α, can be characterized as the invariant distribution of a real Markov chain Γn. In this paper we prove that, if α has finite expectation, the rate of convergence (in total variation) of Γn to Γα is geometric. Upper bounds on the rate of convergence are found which seem effective, especially in the case where α has a support which is not doubly infinite. We use this to study an approximation procedure for \\cal{M}}α, and evaluate the approximation error in simulating \\cal{M}}α using this chain. We include examples for a comparison with some of the existing procedures for approximating \\cal{M}}α, and show that the Markov chain approximation compares well with other methods.