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.

翻译：给定一个平滑函数 $f$，我们提出了一种通用方法，将期望为 $m$ 的蒙特卡洛样本转化为 $f(m)$ 的无偏估计。具体而言，我们构建了基于随机截断 $f$ 的泰勒级数展开并估计截断级数系数的估计器。我们推导了它们的性质，并提出了一种自动设置其调优参数（这些参数依赖于 $m$）的策略，旨在使整个方法易于使用。我们针对特定函数 $f(x)=\log x$ 和 $f(x)=1/x$ 开发了相应方法，因为它们出现在诸如潜变量模型的最大似然估计和非规范化模型的贝叶斯推断等多种统计应用中。我们针对一系列应用进行了详细的数值研究，以评估所提方法的竞争力和可靠性。