On Approximate Convergence Properties of the Gibbs Sampler
Sujit K. Sahu, Faculty of Mathematical Studies, University of Southampton, Highfield, SO17 1BJ, UK.
Gareth O. Roberts, Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF, UK.

## This paper has now appeared as

• Sahu, S. K. and Roberts, G. O. (1999) On Convergence of the EM Algorithm and the Gibbs Sampler. Statistics and Computing. 9, 55--64.
• #### SUMMARY

In this article we investigate the relationship between the EM algorithm and the Gibbs sampler. We show that the approximate rate of convergence of the Gibbs sampler by Gaussian approximation is equal to that of the corresponding EM type algorithm. This helps in implementing either of the algorithms as improvement strategies for one algorithm can be directly transported to the other. In particular, by running the EM algorithm we know approximately how many iterations are needed for convergence of the Gibbs sampler. We also obtain a result that under {\em certain} conditions, the EM algorithm used for finding the maximum likelihood estimates can be slower to converge than the corresponding Gibbs sampler for Bayesian inference. We illustrate our results in a number of realistic examples all based on the generalized linear mixed models.

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S.K.Sahu@maths.soton.ac.uk