Two adaptive rejection sampling schemes for probability density functions log-convex tails

TL;DR

This paper introduces two adaptive rejection sampling algorithms capable of efficiently sampling from a wide range of univariate probability densities, including non-log-concave, multimodal, and tail-convex distributions, expanding the applicability of rejection sampling.

Contribution

The paper proposes two novel adaptive rejection sampling schemes that work with non-log-concave, multimodal, and tail-convex distributions, overcoming limitations of standard ARS methods.

Findings

Algorithms effectively sample from complex distributions.

Acceptance rates improve with each rejected sample.

Numerical examples demonstrate broad applicability.

Abstract

Monte Carlo methods are often necessary for the implementation of optimal Bayesian estimators. A fundamental technique that can be used to generate samples from virtually any target probability distribution is the so-called rejection sampling method, which generates candidate samples from a proposal distribution and then accepts them or not by testing the ratio of the target and proposal densities. The class of adaptive rejection sampling (ARS) algorithms is particularly interesting because they can achieve high acceptance rates. However, the standard ARS method can only be used with log-concave target densities. For this reason, many generalizations have been proposed. In this work, we investigate two different adaptive schemes that can be used to draw exactly from a large family of univariate probability density functions (pdf's), not necessarily log-concave, possibly multimodal and with tails of arbitrary concavity. These techniques are adaptive in the sense that every time a candidate sample is rejected, the acceptance rate is improved. The two proposed algorithms can work properly when the target pdf is multimodal, with first and second derivatives analytically intractable, and when the tails are log-convex in a infinite domain. Therefore, they can be applied in a number of scenarios in which the other generalizations of the standard ARS fail. Two illustrative numerical examples are shown.