Novel schemes for adaptive rejection sampling

Resumen

We address the problem of generating random samples from a target probability distribution, with density function ??, using accept/reject methods. An "accept/reject sampler" (or, simply, a "rejection sampler") is an algorithm that first draws a random variate from a proposal distribution with density ? (where ? ? ??, in general) and then performs a test to determine whether the variate can be accepted as a sample from the target distribution or not. If we apply the algorithm repeatedly until we accept ? times, then we obtain a collection of ? independent and identically distributed (i.i.d.) samples from the distribution with density ??. The goal of the present work is to devise and analyze adaptive rejection samplers that can be applied to generate i.i.d. random variates from the broadest possible class of probability distributions. Adaptive rejection sampling algorithms typically construct a sequence of proposal functions ??, ??,... ??..., such that (a) it is easy to draw i.i.d. samples from them and (b) they converge, in some way, to the density ?? of the target probability distribution. When surveying the literature, it is simple to identify several such methods but virtually all of them present severe limitations in the class of target densities,??, for which they can be applied. The "standard" adaptive rejection sampler by Gilks and Wild, for instance, only works when ?? is strictly log-concave. Through Chapters 3, 4 and 5 we introduce a new methodology for adaptive rejection sampling that can be used with a broad family of target probability densities (including, e.g., multimodal functions) and subsumes Gilks and Wild's method as a particular case. We discuss several variations of the main algorithm that enable, e.g., sampling from some particularly "difficult" distributions (for instance, cases where ?? has log-convex tails and in nite support) or yield "automatic" software implementations using little analytical information about the target density ??. Several numerical examples, including comparisons with some of the most relevant techniques in the literature, are also shown in Chapter 6. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------