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Abstract:
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In this work, we offer a rigorous theoretical treatment to show the role of shared hyperparameters in a hierarchical Bayes model on borrowing of strength or information and their effects on posterior inference. We consider a non-asymptotic framework where observations are sampled from a mixed-effects model and the true joint distribution of the random effects is Gaussian having an equi-correlation structure. We model these effects using a pair of nested hierarchical priors. Defining an integrated l_2 risk function (w.r.t. the true joint distribution of the random effects), we obtain the effect of information borrowing through the corresponding gain in risk. In particular, we find both necessary and sufficient conditions under which the Bayes estimator obtained from the model with a greater degree of hierarchy outperforms its competitor. These conditions ensure that the former will always have an edge over the other except for cases when the hierarchical model corresponding to the latter is sufficiently closer to the truth. However, in the present scenario, occurrence of such situations has very limited significance from both theoretical and practical standpoints.
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