Bayesian methodology for combining the results from different experiments when the specifications for pooling are uncertain

Abstract

Given data from several experiments or observational studies initially believed to be similar, it is desired to estimate the means corresponding to one or more experiments of particular interest. To illustrate the proposed methodology we consider a simple specification where there are no covariates. A class of prior distributions for the set, μ = (μ1,…, μL), of experiment means is specified to reflect the beliefs that (a) there are subsets of μ such that the μ1's within each subset are similar, and (b) the composition of such subsets of μ is uncertain. Such a specification leads to an estimator of μ1 that exhibits the ‘gaining of strength’. However, the nature and amount of the pooling of data from other experiments depends on the observed sample data. We present and discuss the posterior mean and covariance matrix of μ. Both proper and improper prior distributions are considered.