Binary Mixed Effects Models and Improper Priors with Application in Meta-Analysis

The most crucial part in Bayesian analysis is the choice of prior distribution. Improper priors are often used in hierarchical Bayesian models due to the lack of information on the hyper parameters at the lower levels of the hierarchy. When improper priors are used, it is important to establish the posterior propriety. Binary random/mixed effects models are commonly used in Meta analyses of binary outcome data. For severely sparse data the likelihood based estimates, obtained from such models, may tend towards the boundary, and this may hamper Bayesian computation and inference even under proper priors. We establish conditions for posterior propriety for such models. The random effects model we consider includes both parameters of interest and nuisance parameters, and the notion of posterior propriety in this model is linked to the notion of polyhedral cones and the idea of complete and quasi-complete separation in logistic regression. We further illustrate that, even in cases when the prior is diffuse, the Markov chain based computations and Bayesian inference may get adversely affected due to these limiting cases.