The New England Journal of Statistics in Data Science

We consider the class of inverse probability weight (IPW) estimators, including the popular Horvitz–Thompson and Hájek estimators used routinely in survey sampling, causal inference and for Bayesian computation. We focus on the ‘weak paradoxes’ for these estimators due to two counterexamples by Basu (1988) and Wasserman (2004) and investigate the two natural Bayesian answers to this problem: one based on binning and smoothing: a ‘Bayesian sieve’ and the other based on a conjugate hierarchical model that allows borrowing information via exchangeability. We compare the mean squared errors for the two Bayesian estimators with the IPW estimators for Wasserman’s example via simulation studies on a broad range of parameter configurations. We also prove posterior consistency for the Bayes estimators under missing-completely-at-random assumption and show that it requires fewer assumptions on the inclusion probabilities. We also revisit the connection between the different problems where improved or adaptive IPW estimators will be useful, including survey sampling, evidence estimation strategies such as Conditional Monte Carlo, Riemannian sum, Trapezoidal rules and vertical likelihood, as well as average treatment effect estimation in causal inference.