On a Likelihood Approach for Monte Carlo Integration
Use of estimating equations has been a common approach for constructing a Monte Carlo estimator. In an alternative approach, Kong et al. formulate Monte Carlo integration as a statistical model using simulated observations as data. The baseline measure is estimated by maximum likelihood and then integrals of interest are estimated by substituting the estimated measure. For two different situations where independent observations are simulated from multiple distributions, we show that the likelihood approach achieves the lowest asymptotic variance possible by using estimating equations. In the first situation, the normalizing constants of the design distributions need to be estimated, and Meng and Wong's bridge sampling estimator is considered. In the second situation, the values of the normalizing constants are known, thereby imposing linear constraints on the baseline measure. Estimating equations including Hesterberg's stratified importance sampling estimator, Veach and Guibas's multiple importance sampling estimator, and Owen and Zhou's method of control variates are considered.