Towards a turnkey approach for unbiased Monte Carlo estimation of smooth functions of expectations

Abstract Given a smooth function $ f $, we develop a general approach to turn Monte Carlo samples with expectation $ m $ into an unbiased estimate of $ f(m) $. Specifically, we develop estimators that are based on randomly truncating the Taylor series expansion of $ f $ and estimating the coefficients of the truncated series. We derive their properties and propose a strategy to set their tuning parameters (which depend on $ m $) automatically, with a view to making the whole approach simple to use. We develop our methods for the specific functions $ f(x)=\log x $ and $ f(x)=1/x $, as they arise in several statistical applications such as maximum likelihood estimation of latent variable models and Bayesian inference for unnormalized models. Detailed numerical studies are performed for a range of applications to determine how competitive and reliable the proposed approach is.