Multiparameter Bernoulli Factories

TL;DR

This paper characterizes which functions of multiple unknown-biased coins can be sampled from, extending combinatorial sampling methods to the boundary of the hypercube, with implications for complex probabilistic sampling.

Contribution

It provides a complete characterization of functions for which sampling from unknown-biased coins is possible, extending existing sampling procedures to boundary cases.

Findings

Complete characterization of feasible functions for Bernoulli factories.

Extension of Sampford Sampling to the boundary of the hypercube.

Framework for sampling from functions of multiple unknown biases.

Abstract

We consider the problem of computing with many coins of unknown bias. We are given samples access to $n$ coins with \emph{unknown} biases $p_1,\dots, p_n$ and are asked to sample from a coin with bias $f(p_1, \dots, p_n)$ for a given function $f:[0,1]^n \rightarrow [0,1]$. We give a complete characterization of the functions $f$ for which this is possible. As a consequence, we show how to extend various combinatorial sampling procedures (most notably, the classic Sampford Sampling for $k$-subsets) to the boundary of the hypercube.