Perfect simulation from unbiased simulation

Abstract

We show that any application of the technique of unbiased simulation becomes perfect simulation when coalescence of the two coupled Markov chains can be practically assured in advance. This happens when a fixed number of iterations is high enough that the probability of needing any more to achieve coalescence is negligible; we suggest a value of $10^{-20}$. This finding enormously increases the range of problems for which perfect simulation, which exactly follows the target distribution, can be implemented. We design a new algorithm to make practical use of the high number of iterations by producing extra perfect sample points with little extra computational effort, at a cost of a small, controllable amount of serial correlation within sample sets of about 20 points. Different sample sets remain completely independent. The algorithm includes maximal coupling for continuous processes, to bring together chains that are already close. We illustrate the methodology on a simple, two-state Markov chain and on standard normal distributions up to 20 dimensions. Our technical formulation involves a nonzero probability, which can be made arbitrarily small, that a single perfect sample point may have its place taken by a "string" of many points which are assigned weights, each equal to $\pm 1$, that sum to~$1$. A point with a weight of $-1$ is a "hole", which is an object that can be cancelled by an equivalent point that has the same value but opposite weight $+1$.