Multiparameter Bernoulli factories

Abstract

We consider the problem of computing with many coins of unknown bias. We are given access to samples of n coins with unknown biases ${\mathit{p}}_1,\dots ,{\mathit{p}}_{\mathit{n}}$ and are asked to sample from a coin with bias $\mathit{f}({\mathit{p}}_1,\dots ,{\mathit{p}}_{\mathit{n}})$ for a given function $\mathit{f}:{[0,1]}^{\mathit{n}}\to [0,1]$. 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.