Inverse Probability Weighting: the Missing Link between Survey Sampling and Evidence Estimation

We consider the class of inverse probability weight (IPW) estimators, including the popular Horwitz-Thompson and Hajek estimators used routinely in survey sampling, causal inference and evidence estimation 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 show that the two Bayesian estimators achieve lower mean squared errors in Wasserman's example compared to simple IPW estimators via simulation studies on a broad range of parameter configurations. We prove posterior consistency for the Bayes estimator and show how 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.