  This paper considers properties of an optimization based sampler for
targeting the posterior distribution when the likelihood is intractable and
auxiliary statistics are used to summarize information in the data. Our reverse
sampler approximates the likelihood-based posterior distribution by solving a
sequence of simulated minimum distance problems. By a change of variable
argument, these estimates are reweighted with a prior and the volume of the
jacobian matrix to serve as draws from the desired posterior distribution. The
sampler provides a conceptual framework to understand the difference between
two types of likelihood free estimation. Because simulated minimum distance
estimation always results in acceptable draws, the reverse sampler is
potentially an alternative to existing approximate Bayesian methods that are
computationally demanding because of a low acceptance rate.
