Combinatorial Bernoulli Factories: Matchings, Flows and Other Polytopes

A Bernoulli factory is an algorithmic procedure for exact sampling of certain random variables having only Bernoulli access to their parameters. Bernoulli access to a parameter $p \in [0,1]$ means the algorithm does not know $p$, but has sample access to independent draws of a Bernoulli random variable with mean equal to $p$. In this paper, we study the problem of Bernoulli factories for polytopes: given Bernoulli access to a vector $x\in \mathcal{P}$ for a given polytope $\mathcal{P}\subset [0,1]^n$, output a randomized vertex such that the expected value of the $i$-th coordinate is \emph{exactly} equal to $x_i$. For example, for the special case of the perfect matching polytope, one is given Bernoulli access to the entries of a doubly stochastic matrix $[x_{ij}]$ and asked to sample a matching such that the probability of each edge $(i,j)$ be present in the matching is exactly equal to $x_{ij}$. We show that a polytope $\mathcal{P}$ admits a Bernoulli factory if and and only if $\mathcal{P}$ is the intersection of $[0,1]^n$ with an affine subspace. Our construction is based on an algebraic formulation of the problem, involving identifying a family of Bernstein polynomials (one per vertex) that satisfy a certain algebraic identity on $\mathcal{P}$. The main technical tool behind our construction is a connection between these polynomials and the geometry of zonotope tilings. We apply these results to construct an explicit factory for the perfect matching polytope. The resulting factory is deeply connected to the combinatorial enumeration of arborescences and may be of independent interest. For the $k$-uniform matroid polytope, we recover a sampling procedure known in statistics as Sampford sampling.Comment: 41 pages, 9 figure

Exact sampling for intractable probability distributions via a Bernoulli factory

Many applications in the field of statistics require Markov chain Monte Carlo methods. Determining appropriate starting values and run lengths can be both analytically and empirically challenging. A desire to overcome these problems has led to the development of exact, or perfect, sampling algorithms which convert a Markov chain into an algorithm that produces i.i.d. samples from the stationary distribution. Unfortunately, very few of these algorithms have been developed for the distributions that arise in statistical applications, which typically have uncountable support. Here we study an exact sampling algorithm using a geometrically ergodic Markov chain on a general state space. Our work provides a significant reduction to the number of input draws necessary for the Bernoulli factory, which enables exact sampling via a rejection sampling approach. We illustrate the algorithm on a univariate Metropolis-Hastings sampler and a bivariate Gibbs sampler, which provide a proof of concept and insight into hyper-parameter selection. Finally, we illustrate the algorithm on a Bayesian version of the one-way random effects model with data from a styrene exposure study.Comment: 28 pages, 2 figure

A Practical Implementation of the Bernoulli Factory

The Bernoulli Factory is an algorithm that takes as input a series of i.i.d. Bernoulli random variables with an unknown but fixed success probability $p$, and outputs a corresponding series of Bernoulli random variables with success probability $f(p)$, where the function $f$ is known and defined on the interval $[0,1]$. While several practical uses of the method have been proposed in Monte Carlo applications, these require an implementation framework that is flexible, general and efficient. We present such a framework for functions that are either strictly linear, concave, or convex on the unit interval using a series of envelope functions defined through a cascade, and show that this method not only greatly reduces the number of input bits needed in practice compared to other currently proposed solutions for more specific problems, and is easy to specify for simple forms, but can easily be coupled to asymptotically efficient methods to allow for theoretically strong results.Comment: 23 page