Approximate Bayesian inference for random effects meta-analysis

Whilst meta-analysis is becoming a more commonplace statistical technique, Bayesian inference in meta-analysis requires complex computational techniques to be routinely applied. We consider simple approximations for the first and second moments of the parameters of a Bayesian random effects model for meta-analysis. These computationally inexpensive methods are based on simple analytical formulae that provide an efficient tool for a qualitative analysis and a quick numerical estimation of posterior quantities. They are shown to lead to sensible approximations in two examples of meta-analyses and to be in broad agreement with the more computationally intensive Gibbs sampling.