Plummer, Martyn; Best, Nicky; Cowles, Kate and Vines, Karen
CODA: convergence diagnosis and output analysis for MCMC.
R News, 6(1) pp. 7–11.
[1st paragraph] At first sight, Bayesian inference with Markov Chain Monte Carlo (MCMC) appears to be straightforward. The user defines a full probability model, perhaps using one of the programs discussed in this issue; an underlying sampling engine takes the model definition and returns a sequence of dependent samples from the posterior distribution of the model parameters, given the supplied data. The user can derive any summary of the posterior distribution from this sample. For example, to calculate a 95% credible interval for a parameter α, it suffices to take 1000 MCMC iterations of α and sort them so that α1<α2<...<α1000. The credible interval estimate is then (α25, α975). However, there is a price to be paid for this simplicity. Unlike most numerical methods used in statistical inference, MCMC does not give a clear indication of whether it has converged. The underlying Markov chain theory only guarantees that the distribution of the output will converge to the posterior in the limit as the number of iterations increases to infinity. The user is generally ignorant about how quickly convergence occurs, and therefore has to fall back on post hoc testing of the sampled output. By convention, the sample is divided into two parts: a “burn in” period during which all samples are discarded, and the remainder of the run in which the chain is considered to have converged sufficiently close to the limiting distribution to be used. Two questions then arise: 1. How long should the burn in period be? 2. How many samples are required to accurately estimate posterior quantities of interest? The coda package for R contains a set of functions designed to help the user answer these questions. Some of these convergence diagnostics are simple graphical ways of summarizing the data. Others are formal statistical tests.
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