Generalized Poststratification and Importance Sampling for Subsampled Markov Chain Monte Carlo Estimation
Benchmark estimation is motivated by the goal of producing an approximation to a posterior distribution that is better than the empirical distribution function. This is accomplished by incorporating additional information into the construction of the approximation. We focus here on generalized poststratification, the most successful implementation of benchmark estimation in our experience. We develop generalized poststratification for settings where the source of the simulation differs from the posterior that is to be approximated. This allows us to use the techniques in settings where it is advantageous to draw from a distribution different than the posterior, whether for exploration of the data and/or model, for algorithmic simplicity, for improved convergence of the simulation, or for improved estimation of selected features of the posterior. We develop an asymptotic (in simulation size) theory for the estimators, providing conditions under which central limit theorems hold. The central limit theorems apply both to an importance sampling context and to direct sampling from the posterior distribution. The asymptotic results, coupled with large-sample (size of data) approximation results provide guidance on how to implement generalized poststratification. The theoretical results also explain the gains associated with generalized poststratification and the empirically observed robustness to cutpoints for the strata. We note that the results apply well beyond the setting of Markov chain Monte Carlo simulation. The technique is illustrated with an infinite-dimensional semiparametric Bayesian regression model and a low-dimensional, overdispersed hierarchical Bayesian model. In both cases, the technique shows substantial benefits.