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.

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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.
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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