Laplace expansion for posterior densities of nonlinear functions of parameters

SUMMARY: An asymptotic expansion for marginal posterior densities of nonlinear functions of parameters is derived. The density in question is proportional to an integral over a lower dimensional manifold with the integrand including an appropriate change-in-volume term, Laplace's method is applied to obtain the expansion. Avoiding the difficult step of the explicit parameterization of the manifold, we carry out the expansion via an implicit and local parameterization, and the derivatives required can be evaluated recursively. An example indicates a considerable improvement by including the term of order O(n^-1).