Error bounds for multidimensional Laplace approximation

A numerical estimate is obtained for the error associated with the Laplace approximation of the double integral I(λ) = ∝∝_D g(x,y) e^−λf(x,y) dx dy, where D is a domain in , λ is a large positive parameter, f(x, y) and g(x, y) are real-valued and sufficiently smooth, and ∝(x, y) has an absolute minimum in D. The use of the estimate is illustrated by applying it to two realistic examples. The method used here applies also to higher dimensional integrals.