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Bayesian methods for variance component models
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DOI：10.2307/2291669.png)
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Abstract
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An exact Bayesian analysis can be performed for normal theory variance components models, using importance sampling. But remarkably accurate approximations are available using the Laplacian T-approximation introduced by Leonard, Hsu, and Ritter. Instead of maximizing the joint posterior density, conditional upon the parameter of interest, a device described by O'Hagan is used first. The Bayesian estimators are compared to the Lindley-Stein shrinkage estimators and the Lindley-Smith joint modal estimators. It is confirmed that joint modes can overcollapse toward prior hypotheses, when compared with more sensible Bayesian procedures. This is referred to as a ''collapsing phenomenon.'' A numerical example from the one-way random-effects model is considered, and the risks of the different estimators are simulated under a variety of loss functions. It is concluded that although the Lindley-Stein estimator performs well, a full hierarchical Bayesian analysis performs at least equally well, while permitting more detailed finite-sample inference regarding any parameter of interest.
Keywords:
analysis of variance
empirical Bayes
hierarchical Bayes
importance sampling
joint mode
Laplacian T approximation
marginal mode
maximum component loss
risk function
shrinkage estimator
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