A debiased Bernoulli factory and unbiased estimation of a probability

Given a known function f : [0, 1] ↦ (0, 1) and a random but almost surely finite number of independent, Ber$(x)$-distributed random variables with unknown x ∈ [0, 1], we construct an unbiased, $[0, 1]$-valued estimator of the probability f(x) ∈ (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.