Generalized laplace approximation and its application to credibility theory

초록

Laplace approximation provides a Gaussian approximation of a posterior distribution via a second-order Taylor expansion. Although the Bernstein-von Mises theorem guarantees asymptotic normality as the sample size approaches infinity, the Gaussian approximation may be unreliable when the sample size is finite. This is particularly true when the posterior distribution is skewed, which is a common occurrence in the insurance ratemaking process, where the use of a Gaussian distribution may not yield an accurate approximation. In this study, by utilizing the generalized version of Taylor expansion [Widder (1928). A generalization of Taylor's series. Transactions of the American Mathematical Society, 30(1), 126-154], we introduce a generalized version of Laplace approximation where the posterior distribution is approximated by various parametric distributions in the exponential family. We apply this method to random effects models, connecting it to credibility premium, in the insurance context. While credibility premium provides an affine posterior mean approximation, it lacks further distributional information. Our method introduces the ability to approximate the posterior distribution, while still providing the same point approximation as credibility premium. Numerical analysis confirms the effectiveness of the proposed approach.

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