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Empirical Bayes methods can be seen as an approximation to a fully Bayesian treatment of a hierarchical Bayes model.. In, for example, a two-stage hierarchical Bayes model, observed data = {,, …,} are assumed to be generated from an unobserved set of parameters = {,, …,} according to a probability distribution ().
Best linear unbiased predictions are similar to empirical Bayes estimates of random effects in linear mixed models, except that in the latter case, where weights depend on unknown values of components of variance, these unknown variances are replaced by sample-based estimates.
A later paper by Copas [3] applies shrinkage in a context where the problem is to predict a binary response on the basis of binary explanatory variables. Hausser and Strimmer "develop a James-Stein-type shrinkage estimator, resulting in a procedure that is highly efficient statistically as well as computationally.
The sub-models combine to form the hierarchical model, and Bayes' theorem is used to integrate them with the observed data and account for all the uncertainty that is present. The result of this integration is it allows calculation of the posterior distribution of the prior , providing an updated probability estimate.
Bayesian model reduction was subsequently generalised and applied to other forms of Bayesian models, for example parametric empirical Bayes (PEB) models of group effects. [2] Here, it is used to compute the evidence and parameters for any given level of a hierarchical model under constraints (empirical priors) imposed by the level above.
For large samples, the shrinkage intensity will reduce to zero, hence in this case the shrinkage estimator will be identical to the empirical estimator. Apart from increased efficiency the shrinkage estimate has the additional advantage that it is always positive definite and well conditioned. Various shrinkage targets have been proposed:
A Bayes estimator derived through the empirical Bayes method is called an empirical Bayes estimator. Empirical Bayes methods enable the use of auxiliary empirical data, from observations of related parameters, in the development of a Bayes estimator. This is done under the assumption that the estimated parameters are obtained from a common prior.
The computation of Bayes factors on summary statistics may not be related to the Bayes factors on the original data, which may therefore render the results meaningless. Only use summary statistics that fulfill the necessary and sufficient conditions to produce a consistent Bayesian model choice.