Consistency of Importance Sampling estimates based on dependent sample sets and an application to models with factorizing likelihoods

In this paper, I proof that Importance Sampling estimates based on dependent sample sets are consistent under certain conditions. This can be used to reduce variance in Bayesian Models with factorizing likelihoods, using sample sets that are much larger than the number of likelihood evaluations, a technique dubbed Sample Inflation. I evaluate Sample Inflation on a toy Gaussian problem and two Mixture Models.