Multiparameter bernoulli factories

We consider the problem of computing with many coins of unknown bias. We are given access to samples of n coins with unknown biases p1, . . . , pn and are asked to sample from a coin with bias f (p1, ... , pn) for a given function f : [0, 1]n -> [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.