Generalized Laplacian approximations in Bayesian inference

This paper presents a new Laplacian approximation to the posterior density of eta = g(theta). It has a simpler analytical form than that described by Leonard et al. (1989). The approximation derived by Leonard et al. requires a conditional information matrix R(eta) to be positive definite for every fixed eta. However, in many cases, not all R(eta) are positive definite. In such cases, the computations of their approximations fail, since the approximation cannot be normalized. However, the new approximation may be modified so that the corresponding conditional information matrix can be made positive definite for every fixed eta. In addition, a Bayesian procedure for contingency-table model checking is provided. An example of cross-classification between the educational level of a wife and fertility-planning status of couples is used for explanation. Various Laplacian approximations are computed and compared in this example and in an example of public school expenditures in the context of Bayesian analysis of the multiparameter Fisher-Behrens problem.