A debiased Bernoulli factory and unbiased estimation of a probability
Jere Koskela, Toni Karvonen, Krzysztof Łatuszyński, Dario Spanò
TL;DR
This paper addresses constructing unbiased $[0,1]$-valued estimators of $f(x)$ from an unknown $x\in[0,1]$ using a finite number of Ber$(x)$ inputs by a debiasing strategy that truncates a telescoping series of consistent estimators derived from a Bernoulli factory for $f$. The main result shows that, for $f\in C^{\rho}[0,1]$ with $\rho>5$, one can choose parameters so that the debiased estimator $\psi(L,S_L)$ is unbiased, nonnegative, and almost surely lies in $[0,1]$, with $\mathbb{E}[\psi(L,S_L)]=f(x)$ for all $x$ and the required input-coin count independent of their outcomes. The construction leverages polynomial-approximation bounds (via $g_n,h_n$ and coefficients $a(n,k),b(n,k)$) to produce a constructive $f$-factory with near-optimal tail properties, and yields corollaries on nonnegativity, an upper bound by $1$, and a variance bound of $1/4$. The discussion highlights efficiency considerations, conjectures exponential tails for highly smooth $f$, and outlines practical challenges and connections to related Bernoulli factory schemes in the literature. Its approach advances unbiased estimation in Bernoulli-factory contexts and suggests potential applications in debiased Monte Carlo and pseudo-marginal methods.
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.