Experiences with Approximate Bayes Inference for the Poisson-CAR Model

In many areas of epidemiologic, demographic and geographical research, inference based on hierarchical spatial regression models is popular and important; for example, in disease mapping, environmental and health monitoring studies. Several estimation and inferential procedures have been proposed for these models, utilizing a variety of methods such as estimating equations, empirical Bayes and hierarchical Bayes. Hierarchical Bayes provides the full range of statistical inference (point as well as interval estimation) which may not be readily available in the other approaches. However, hierarchical Bayes is not problem-free and computations can be challenging in complex applications. Recently, an alternative method, namely the approximate Bayes, has been proposed to alleviate the problems with the hierarchical Bayes method. Approximate Bayes uses an integrated nested Laplace approximation to derive numerical approximations to various marginals of the full posterior distributions, thus avoiding Markov Chain Monte Carlo sampling completely. In this article, we compare and contrast between approximate Bayes, hierarchical Bayes and two other inferential methodologies in the context of hierarchical spatial regression models. Our emphasis is to investigate some of the claims made on approximate Bayes, namely the computational gain and the extent of automation, in the implementation. The differences have been demonstrated via simulation as well as through real examples.