  The frequentist method of simulated minimum distance (SMD) is widely used in
economics to estimate complex models with an intractable likelihood. In other
disciplines, a Bayesian approach known as Approximate Bayesian Computation
(ABC) is far more popular. This paper connects these two seemingly related
approaches to likelihood-free estimation by means of a Reverse Sampler that
uses both optimization and importance weighting to target the posterior
distribution. Its hybrid features enable us to analyze an ABC estimate from the
perspective of SMD. We show that an ideal ABC estimate can be obtained as a
weighted average of a sequence of SMD modes, each being the minimizer of the
deviations between the data and the model. This contrasts with the SMD, which
is the mode of the average deviations. Using stochastic expansions, we provide
a general characterization of frequentist estimators and those based on
Bayesian computations including Laplace-type estimators. Their differences are
illustrated using analytical examples and a simulation study of the dynamic
panel model.
