Discussion Of Monte Carlo Simulation For Computing Normalizing Constants

Computing (ratios of) normalizing constants of probability models is a fundamental computational problem for many statistical and scientific studies.There are three common approaches for dealing with this problem :(1) analytic approximation, (2)numerical integration,(3)Monte Carlo simulation.Among them,Monte Carlo simulation is widely used arid especially efficient for dealing with complex,high-dimensional models.Since the 1970's,there have been many Monte Carlo ways for simulating normalizing constants.In this paper,we first give an introduction of importance sampling,bridge sampling ,path sampling and their applications in theoretical physics( thermodynarnic integration) and numerical analysis (Ogata's method).Among them, path sampling has the least Monte Carlo error. Then we discuss the choice of optimal path in path sampling ,which turns out to have close connection with the Jeffreys prior density and the Rao and Hellinger distances between two densities.One of important applicatons of this problem in statistics is model selection by using Bayes factor.At the last part of this paper,according to [7],we discuss how to compute Bayes factor with path sampling for nonlinear structural equations with fixed covariates.Furthermore ,we generalize current nonlinear model to allow nonlinearity of the latent variables in the measurement equation. We also provide simulation studies for the above two cases using softwareWinBUGS,and find that path sampling is an excellent tool in computing Bayes factor for model selectin.