Quantification of Empirical Determinacy: The Impact of Likelihood Weighting on Posterior Location and Spread in Bayesian Meta-Analysis Estimated with JAGS and INLA

The popular Bayesian meta-analysis expressed by the normal-normal hierarchical model synthesizes knowledge from several studies and is highly relevant in practice. The normal-normal hierarchical model is the simplest Bayesian hierarchical model, but illustrates problems typical in more complex Bayesian hierarchical models. Until now, it has been unclear to what extent the data determines the marginal posterior distributions of the parameters in the normal-normal hierarchical model. To address this issue we computed the second derivative of the Bhattacharyya coefficient with respect to the weighted likelihood. This quantity, which we define as the total empirical determinacy (TED), can be written as the sum of two terms: the empirical determinacy of location (EDL), and the empirical determinacy of spread (EDS). We implemented this method in the R package ed4bhm and considered two case studies and one simulation study. We quantified TED, EDL and EDS under different modeling conditions such as model parametrization, the primary outcome, and the prior. This clarifies to what extent the location and spread of the marginal posterior distributions of the parameters are determined by the data. Although these investigations focused on Bayesian normal-normal hierarchical model, the method proposed is applicable more generally to complex Bayesian hierarchical models.