Bayesian statistical inference for sampling from weighted distribution models is studied. Small-sample Bayesian bootstrap clone (BBC) approximations to the posterior distribution are discussed. A second-order property for the BBC in unweighted i.i.d. sampling is given. A consequence is that BBC approximations to a posterior distribution of the mean and to the sampling distribution of the sample average, can be made asymptotically accurate by a proper choice of the random variables that generate the clones. It also follows from this result that in weighted sampling models, BBC approximations to a posterior distribution of the reciprocal of the weighted mean are asymptotically accurate; BBC approximations to a sampling distribution of the reciprocal of the empirical weighted mean are also asymptotically accurate.
"A Bayesian Method for Weighted Sampling." Ann. Statist. 21 (4) 2138 - 2148, December, 1993. https://doi.org/10.1214/aos/1176349414