Joint Statistics / Biostatistics Seminar series
Statistical Models for Monte Carlo Integration
Zhiqiang Tan, University of Chicago
Statistics / Biostatistics and Medical Informatics Assistant Professor Candidate
Thursday, February 20, 2003, 4-5 p.m.
1221 Computer Science and Statistics Center, 1210 W. Dayton St.
Nontrivial integration arises in many statistical analyses, like evaluating the likelihood for spatial models, mixed-effect models, missing data problems, and genetic pedigree analysis, and calculating the Bayes factor for Bayesian model selection. In this talk, I will show a novel approach which formulates Monte Carlo integration as a statistical model using simulated observations as data, and demonstrate the soundness and usefulness of the approach.
The key idea of the approach is to ignore part of the information about the baseline measure and treat the measure as a parameter in a semiparametric model, which is estimated by maximum likelihood. The integrals of interest are estimated as linear functionals. This formulation is of theoretical importance because it provides a unified framework to study Monte Carlo estimation procedures. I will show several results about the maximum likelihood estimator and its comparison to other estimators. The formulation is also of practical importance because it provides an effective way to construct new Monte Carlo estimators. I will show a new method to estimate integrals by iterative simulation. For integrands proportional to the stationary density, the estimator converges at a rate faster than the square root rate.
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