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

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 make 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 un-normalised models. Detailed numerical studies are performed for a range of applications to determine how competitive and reliable the proposed approach is.