On the Generalized Ratio of Uniforms as a Combination of Transformed Rejection and Extended Inverse of Density Sampling

TL;DR

This paper explores the relationships among classical sampling methods like inverse density, transformed rejection, and generalized ratio of uniforms, revealing their equivalences and extending their applicability to non-monotonic and unbounded densities.

Contribution

It demonstrates the equivalence between GRoU and transformed rejection methods for monotonic densities and extends GRoU to handle non-monotonic and unbounded densities.

Findings

GRoU can be derived from transformed rejection for monotonic PDFs

The inverse density method is equivalent to GRoU for monotonic densities

A new GRoU technique is proposed for unbounded densities

Abstract

In this work we investigate the relationship among three classical sampling techniques: the inverse of density (Khintchine's theorem), the transformed rejection (TR) and the generalized ratio of uniforms (GRoU). Given a monotonic probability density function (PDF), we show that the transformed area obtained using the generalized ratio of uniforms method can be found equivalently by applying the transformed rejection sampling approach to the inverse function of the target density. Then we provide an extension of the classical inverse of density idea, showing that it is completely equivalent to the GRoU method for monotonic densities. Although we concentrate on monotonic probability density functions (PDFs), we also discuss how the results presented here can be extended to any non-monotonic PDF that can be decomposed into a collection of intervals where it is monotonically increasing or decreasing. In this general case, we show the connections with transformations of certain random variables and the generalized inverse PDF with the GRoU technique. Finally, we also introduce a GRoU technique to handle unbounded target densities.