Perfection within Reach: Exact MCMC Sampling

The amount of research done by the MCMC community has been very impressive in the last two decades, as testified by this very volume. The power of MCMC has been demonstrated in countless instances in which more traditional numerical algorithms are helpless. However, one ubiquitous problem remains: the detection of convergence or lack thereof. Among the large number of procedures designed for detecting lack of convergence or for establishing convergence bounds (e.g., see Chapters X and XX in this volume), there is one class of MCMC algorithms that stands apart simply because it avoids the problem altogether. Whereas examples of such algorithms can be traced back to at least 1989 (see [56]), it is Propp and Wilson’s 1996 seminal paper [48] that introduced the general scheme of coupling from the past (CFTP). Since then, there has been an intense search and research for perfect sampling or exact sampling algorithms, so named because such algorithms use Markov chains and yet obtain genuine i.i.d. draws—hence perfect or exact—from their limiting distributions within a finite numbers of iterations.