Given a known function $f : [0, 1] \mapsto (0, 1)$ and a random but almost surely finite number of independent, Ber$(x)$-distributed random variables with unknown $x \in [0, 1]$, we construct an unbiased, $[0, 1]$-valued estimator of the probability $f(x) \in (0, 1)$. Our estimator is based on so-called debiasing, or randomly truncating a telescopic series of consistent estimators. Constructing these consistent estimators from the coefficients of a particular Bernoulli factory for $f$ yields provable upper and lower bounds for our unbiased estimator. Our result can be thought of as a novel Bernoulli factory with the appealing property that the required number of Ber$(x)$-distributed random variates is independent of their outcomes, and also as constructive example of the so-called $f$-factory.

翻译：给定已知函数 $f : [0, 1] \mapsto (0, 1)$ 及一个随机但几乎必然有限的独立 Ber$(x)$ 分布随机变量序列（其中 $x \in [0, 1]$ 未知），我们构造了一个取值于 $[0, 1]$ 区间且无偏的概率 $f(x) \in (0, 1)$ 估计量。该估计量基于所谓的去偏技术，即通过随机截断一致估计量的级数展开实现。通过从针对 $f$ 的特定伯努利工厂系数中构造这些一致估计量，我们获得了无偏估计量的可证明上下界。本研究成果可视为一种新型伯努利工厂，其显著特性在于所需 Ber$(x)$ 分布随机变量的数量与其输出结果无关，同时也为所谓的 $f$-工厂提供了建设性示例。