No. 1031

Title:

Importance sampling method with the estimated sampler

Author(s):

Henmi, Masayuki (Department of Statistics, University of Warwick);
Yoshida, Ryo (Institute of Medical Science, University of Tokyo);
Eguchi, Shinto (The Institute of Statistical Mathematics and Graduate University of Advanced Studies)

Key words:

EM algorithm; Information geometry; Mixture model; Monte Carlo integration; Nuisance parameter

Abstract:

    Monte Carlo importance sampling for evaluating numerical integration is discussed in this paper. We consider a parametric family of sampling distributions, and propose the use of the sampling distribution estimated by maximum likelihood. The proposed method of importance sampling using the estimated sampling distribution is shown to improve the asymptotic variance of the ordinary method using the true sampling distribution. The argument is closely related to discussion of the paradox in Henmi & Eguchi (Biometrika 91, 2004). We focus on a condition under which the estimated integration value obtained by the proposed method has asymptotic zero variance, and higher order asymptotics of the variance and bias are investigated. Furthermore, a more special setting for the target function and the sampling distribution model leads us that the estimated importance sampling gives the exact integration value. We demonstrate the applicability and efficiency of the proposed method through several examples including Bayesian analyses on quadratic logistic model and nonlinear normal model, and likelihood analysis on non-normal random effect model.