On a likelihood approach for Monte Carlo integration

TY - JOUR

T1 - On a likelihood approach for Monte Carlo integration

AU - Tan, Zhiqiang

PY - 2004/12

Y1 - 2004/12

N2 - The use of estimating equations has been a common approach for constructing Monte Carlo estimators. Recently, Kong et al. proposed a formulation of Monte Carlo integration as a statistical model, making explicit what information is ignored and what is retained about the baseline measure. From simulated 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 in which independent observations are simulated from multiple distributions, we show that this likelihood approach achieves the lowest asymptotic variance possible by using estimating equations. In the first situation, the normalizing constants of the design distributions are estimated, and Meng and Wong's bridge sampling estimating equation 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.

AB - The use of estimating equations has been a common approach for constructing Monte Carlo estimators. Recently, Kong et al. proposed a formulation of Monte Carlo integration as a statistical model, making explicit what information is ignored and what is retained about the baseline measure. From simulated 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 in which independent observations are simulated from multiple distributions, we show that this likelihood approach achieves the lowest asymptotic variance possible by using estimating equations. In the first situation, the normalizing constants of the design distributions are estimated, and Meng and Wong's bridge sampling estimating equation 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.

KW - Bridge sampling

KW - Control variate

KW - Importance sampling

KW - Stratified sampling

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U2 - https://doi.org/10.1198/016214504000001664

DO - https://doi.org/10.1198/016214504000001664

M3 - Article

SN - 0162-1459

VL - 99

SP - 1027

EP - 1036

JO - Journal of the American Statistical Association

JF - Journal of the American Statistical Association

IS - 468

ER -