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 10β20. 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 Β±1,
that sum to~1. A point with a weight of β1 is a "hole", which is an object
that can be cancelled by an equivalent point that has the same value but
opposite weight +1.Comment: 17 pages, 4 figures; for associated R scripts, see
https://github.com/George-Leigh/PerfectSimulatio