Bayesian Marginal Inference

Abstract A method is proposed for approximating the marginal posterior density of a continuous function of several unknown parameters, thus permitting inferences about any parameter of interest for nonlinear models when the sample size is finite. Possibly tedious numerical integrations are replaced by conditional maximizations, which are shown to be quite accurate in a number of special cases. There are similarities with the profile likelihood ideas originated by Kalbfleisch and Sprott (1970), and the method is contrasted with a Laplacian approximation recommended by Kass, Tierney, and Kadane (1988, in press), referred to here as the “KTK procedure.” The methods are used to approximate the marginal posterior densities of the log-linear interaction effects and an overall measure of association in a two-way contingency table. Snee's (1974) hair/eye color data are reanalyzed, and adjustments are proposed to Goodman's (1964) analysis for the full-rank interaction model. Another application concerns marginalization problems for a discrete p-parameter exponential family distribution, and inferences are considered for the probability of a zero count.