From the Bernoulli Factory to a Dice Enterprise via Perfect Sampling of Markov Chains

by   Giulio Morina, et al.

Given a p-coin that lands heads with unknown probability p, we wish to produce an f(p)-coin for a given function f: (0,1) → (0,1). This problem is commonly known as the Bernoulli Factory and results on its solvability and complexity have been obtained in <cit.>. Nevertheless, generic ways to design a practical Bernoulli Factory for a given function f exist only in a few special cases. We present a constructive way to build an efficient Bernoulli Factory when f(p) is a rational function with coefficients in R. Moreover, we extend the Bernoulli Factory problem to a more general setting where we have access to an m-sided die and we wish to roll a v-sided one; i.e., we consider rational functions f: Δ^m-1→Δ^v-1 between open probability simplices. Our construction consists of rephrasing the original problem as simulating from the stationary distribution of a certain class of Markov chains - a task that we show can be achieved using perfect simulation techniques with the original m-sided die as the only source of randomness. In the Bernoulli Factory case, the number of tosses needed by the algorithm has exponential tails and its expected value can be bounded uniformly in p. En route to optimizing the algorithm we show a fact of independent interest: every finite, integer valued, random variable will eventually become log-concave after convolving with enough Bernoulli trials.


page 1

page 2

page 3

page 4


Skolem and Positivity Completeness of Ergodic Markov Chains

We consider the following decision problems: given a finite, rational Ma...

Stationarity in the Realizations of the Causal Rate-Distortion Function for One-Sided Stationary Sources

This paper derives novel results on the characterization of the the caus...

An Orthogonality Principle for Select-Maximum Estimation of Exponential Variables

It was recently proposed to encode the one-sided exponential source X in...

Optimal bounds for bit-sizes of stationary distributions in finite Markov chains

An irreducible stochastic matrix with rational entries has a stationary ...

Algebraic solution of tropical best approximation problems

We consider discrete best approximation problems formulated and solved i...

Exact Distributed Sampling

Fast distributed algorithms that output a feasible solution for constrai...

Perfect simulation from unbiased simulation

We show that any application of the technique of unbiased simulation bec...

Please sign up or login with your details

Forgot password? Click here to reset