Application of higher-order approximations in Bayesian inference -

Abstract

In Bayesian methods, one almost is required to calculate certain characteristics of posterior and predictive distributions, including the mean, variance and density. When a conjugate prior likelihood pair is used, calculations of these tasks are usually immediate. However, in most useful applications, it is hard to find conjugate priors and so the posterior calculations cannot be obtained in closed form. In such cases analytic or numerical approximations are then needed. In these cases, it is often useful to have analytic approximations that are more accurate than the usual first order normal approximation but at the same time are not as computationally intensive as numerical integration, especially in cases with high dimensional parameter space. For several particular case studies including single and multi-parameter cases, we explored the use of higher order Laplace approximation in getting such estimates and compared the estimates with those obtained via Monte Carlo Methods. The methods will be illustrated by a genetic linkage model and a censored regression model.