  We consider Bayesian inference by importance sampling when the likelihood is
analytically intractable but can be unbiasedly estimated. We refer to this
procedure as importance sampling squared (IS2), as we can often estimate the
likelihood itself by importance sampling. We provide a formal justification for
importance sampling when working with an estimate of the likelihood and study
its convergence properties. We analyze the effect of estimating the likelihood
on the resulting inference and provide guidelines on how to set up the
precision of the likelihood estimate in order to obtain an optimal tradeoff?
between computational cost and accuracy for posterior inference on the model
parameters. We illustrate the procedure in empirical applications for a
generalized multinomial logit model and a stochastic volatility model. The
results show that the IS2 method can lead to fast and accurate posterior
inference under the optimal implementation.
