Bayesian model selection using exact and approximated posterior probabilities with applications to Star Data

Abstract

This research consists of two parts. The first part examines the posterior probability integrals for a family of linear models which arises from the work of Hart, Koen and Lombard (2003). Applying Laplace's method to these integrals is not entirely straightforward. One of the requirements is to analyze the asymptotic behavior of the information matrices as the sample size tends to infinity. This requires a number of analytic tricks, including viewing our covariance matrices as tending to differential operators. The use of differential operators and their Green's functions can provide a convenient and systematic method to asymptotically invert the covariance matrices. Once we have found the asymptotic behavior of the information matrices, we will see that in most cases BIC provides a reasonable approximation to the log of the posterior probability and Laplace's method gives more terms in the expansion and hence provides a slightly better approximation. In other cases, a number of pathologies will arise. We will see that in one case, BIC does not provide an asymptotically consistent estimate of the posterior probability; however, the more general Laplace's method will provide such an estimate. In another case, we will see that a naive application of Laplace's method will give a misleading answer and Laplace's method must be adapted to give the correct answer. The second part uses numerical methods to compute the "exact" posterior probabilities and compare them to the approximations arising from BIC and Laplace's method.