Reweighting Monte Carlo Mixtures
by Charles J. Geyer
Technical Report No. 568
School of Statistics
University of Minnesota
December 6, 1991
Research supported in part by grant DMS-9007833 from the National Science
Foundation.
Abstract
Markov chain Monte Carlo (e.g., the Metropolis algorithm, Hastings algorithm
and Gibbs
sampler) is a general multivariate simulation method applicable to a wide
range of problem. It permits sampling
from any stochastic process whose density is known up to a constant of
proportionality. The Gibbs sampler has
recently received much attention as a method of simulating from posterior
distributions in Bayesian inference, but Markov chain Monte Carlo is no less
important in frequentist inference with applications in maximum likelihood,
hypothethis testing and the parametric bootstrap. It is most useful when
combined with importance reweighting so that a Monte Carlo sample from one
distribution can be used for inference about many distributions.
In Bayesian inference,
reweighting permits the calculation of posteriors corresponding to a range
of priors using a Monte Carlo sample from just one posterior. In likelihood
inference, reweighting permits the calculation of the whole likelihood
function using a Monte Carlo sample from just one distribution in the model.
Given this estimate of the likelihood, a parametric bootstrap calculation
of the sampling distribution of the maximum likelihood estimate can be done
using just one more Monte Carlo sample.
Although reweighting can save much calculation, it does not work well unless
the distribution being reweighted places appreciable mass in all regions
of interest. Hence it is often not advisable to sample from a distribution
in the model. Reweighting a mixture of distributions in the model may perform
much better. But using such a mixture gives rise to another problem when the
densities are known only up to constants of proportionality. These
normalizing constants must be calculated to obtain the mixture density.
Direct Monte Carlo regression accurately estimates these constants,
permitting the use of these mixture estimates in Markov chain Monte Carlo.