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

Given a smooth function , we develop a general approach to turn Monte Carlo samples with expectationinto an unbiased estimate of . Specifically, we develop estimators that are based on randomly truncating the Taylor series expansion ofand estimating the coefficients of the truncated series. We derive their properties and propose a strategy to set their tuning parameters (which depend on ) automatically, with a view to making the whole approach simple to use. We develop our methods for the specific functionsand , 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.