A debiased Bernoulli factory and unbiased estimation of a probability

Abstract

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.