Laplace Approximation of High Dimensional Integrals
研究了当积分维度与极限参数n相当时，标准拉普拉斯近似失效的问题，并提出了通过重新组合形式展开中的项来构造有效渐近展开的方法。
SUMMARY It is shown that the usual Laplace approximation is not a valid asymptotic approximation when the dimension of the integral is comparable with the limiting parameter n. The formal Laplace expansion for multidimensional integrals is given and used to construct asymptotic approximations for high dimensional integrals. One example is considered in which the dimension of the integral is O(n 1/2) and the relative error of the unmodified Laplace approximation is O(1). Nevertheless, it is possible to construct a valid asymptotic expansion by regrouping terms in the formal expansion according to asymptotic order in n.