Laplace approximations to means and variances with asymptotic modes

The moment-generating function method, which is proposed by Tierney et al. [1989a. Fully exponential Laplace approximations to expectations and variances of nonpositive functions. J. Amer. Statist. Assoc. 84, 710–716], is an asymptotic technique of approximating a posterior mean of a general function by approximating the moment-generating function (MGF), and then differentiating it. In this article, we give approximations to the posterior means and variances by combining the MGF method and the Laplace approximations with asymptotic modes. We prove that asymptotic errors of the approximate means and variances are of order n^-2$n^{-2}$ and of order n^-3$n^{-3}$, respectively. Our approximation is closely related to a standard-form approximation, and is given without evaluating the exact posterior mode and third derivatives of the log-likelihood function. The MGF method also improves numerical instability of the fully exponential Laplace approximation for a predictive mean in logistic regression.