Simulating Events of Unknown Probabilities via Reverse Time Martingales

TL;DR

This paper introduces a versatile framework for simulating events with unknown probabilities using reverse time martingales, applicable to complex problems like diffusions and Bernoulli factories, with practical algorithms and theoretical insights.

Contribution

It develops a general, implementable framework for simulating events of unknown probability, unifying and extending existing algorithms for diffusions and Bernoulli factories.

Findings

Unified algorithms for simulating events with unknown probabilities.

Application to diffusions and Bernoulli factory problems.

Improved understanding and implementation of key simulation algorithms.

Abstract

Assume that one aims to simulate an event of unknown probability $s\in (0,1)$ which is uniquely determined, however only its approximations can be obtained using a finite computational effort. Such settings are often encountered in statistical simulations. We consider two specific examples. First, the exact simulation of non-linear diffusions, second, the celebrated Bernoulli factory problem of generating an $f(p)-$coin given a sequence $X_1,X_2,...$ of independent tosses of a $p-$coin (with known $f$ and unknown $p$). We describe a general framework and provide algorithms where this kind of problems can be fitted and solved. The algorithms are straightforward to implement and thus allow for effective simulation of desired events of probability $s.$ In the case of diffusions, we obtain the algorithm of \cite{BeskosRobertsEA1} as a specific instance of the generic framework developed here. In the case of the Bernoulli factory, our work offers a statistical understanding of the Nacu-Peres algorithm for $f(p) = \min\{2p, 1-2\varepsilon\}$ (which is central to the general question) and allows for its immediate implementation that avoids algorithmic difficulties of the original version.