A simulation approach to convergence rates for Markov chain Monte Carlo algorithms |
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Authors: | MARY KATHRYN COWLES JEFFREY S. ROSENTHAL |
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Affiliation: | (1) Department of Biostatistics, Harvard School of Public Health, Boston, MA 02115, USA;(2) Department of Statistics, University of Toronto, Toronto, Ontario, Canada, M5S 3G3 |
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Abstract: | Markov chain Monte Carlo (MCMC) methods, including the Gibbs sampler and the Metropolis–Hastings algorithm, are very commonly used in Bayesian statistics for sampling from complicated, high-dimensional posterior distributions. A continuing source of uncertainty is how long such a sampler must be run in order to converge approximately to its target stationary distribution. A method has previously been developed to compute rigorous theoretical upper bounds on the number of iterations required to achieve a specified degree of convergence in total variation distance by verifying drift and minorization conditions. We propose the use of auxiliary simulations to estimate the numerical values needed in this theorem. Our simulation method makes it possible to compute quantitative convergence bounds for models for which the requisite analytical computations would be prohibitively difficult or impossible. On the other hand, although our method appears to perform well in our example problems, it cannot provide the guarantees offered by analytical proof. |
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Keywords: | Drift condition Gibbs sampler Markov chain Monte Carlo Metropolis– Hastings algorithm minorization condition ordinal probit variance components |
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