Posterior explorations of Markov chains in Bayesian analysis of discrete finite mixture models
P M Saama · Bulletin - International Bull Evaluation Service/Interbull bulletin · 1999
Markov Chain Monte Carlo (MCMC) methods make possible the use of flexible Bayesian models that would otherwise be computationally infeasible. In essence, MCMC methods involve sampling from a particular posterior distribution by simulating a Markov Chain with that posterior as its stationary density. However, one must decide when to stop the iterations, or more precisely, judge how close the underlying algorithm is to convergence after a specified number of iterations. Furthermore, an MCMC simulation converges to a target distribution, rather than a target point, and inferences are based on moments of that target distribution. Problems in mixtures arise because the mixing distribution is unknown and, in Bayesian nonparametric analysis, it is considered as a random distribution function which is usually given a Dirichlet process prior. This paper examines the performance of an L distance convergence diagnostic which assesses the convergence of the joint density of a Gibbs Sampler algorithm in a discrete finite mixture model. Basically, the convergence diagnostic method measures the difference between distributions of a fixed number of replications sub-sampled from independent Markov chains by (over)estimating the total variation distance between the densities. Initially, the problem of determining the probability that a given data point is assigned to a given component in a mixture is addressed. Then the convergence diagnostic is illustrated and interpreted using simulated data. It is shown, that this diagnostic has advantages over many existing convergence diagnostics in terms of consistency, applicability, computational expense, and interpretability.