Higher asymptotics of Laplace's approximation

Abstract

We present expressions for the coefficients which arise in asymptotic expansions of multiple integrals of Laplace type (the first term of which is known as Laplace's approximation) in terms of asymptotic series of the functions in the integrand. Our most general result assumes no smoothness of the functions of the integrand, but the expressions we obtain contain integrals which may be difficult to evaluate in practice. We then discuss additional assumptions which are sufficient to simplify these integrals, and for the common case that the functions in the integrand admit (finite) Taylor series and the exponent has a nondegenerate minimum, we evaluate the integrals to obtain explicit formulae for the coefficients.