Abstract

This article reviews some second-order approximations to the posterior mean, and the marginal distribution of observations called a marginal likelihood, where “second-order” means that the approximation has asymptotic errors of order O ( n − 2 ) O(n^{-2}) as the sample size n n goes to infinity. In addition, the Laplace approximations using asymptotic modes instead of the exact mode of integrand, and higher-order asymptotic expansions are presented.

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