The Gibbs sampler has proved to be a popular Markovian iterative scheme for sampling from multi-dimensional distributions, especially for applications within Bayesian statistics. Despite the existence of its more general counterpart, the Metropolis-Hastings algorithm, the Gibbs sampler has remained the default option for a wide range of problems. However, the currently available analyses and the convergence assessment methods of the Gibbs sampler are neither foolproof nor very powerful.
This article aims to provide a method for approximately pre-determining convergence properties of the Gibbs sampler. This is to be done by analysing the convergence properties of the Gibbs sampler on a normal approximation of the target distribution. In general, the limiting convergence properties of the Gibbs samplers on a sequence of target distribution (approaching a limit) are not the same as the convergence properties of the Gibbs sampler on the limiting target distribution. Theoretical results are given in this article to justify that under conditions, the convergence properties of the Gibbs sampler can be approximated as well. A number of practical examples are given for illustration.
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