Grupo de Tratamiento de Señal y Aprendizaje
http://hdl.handle.net/10016/9041
2015-10-08T18:05:40ZTwo adaptive rejection sampling schemes for probability density functions log-convex tails
http://hdl.handle.net/10016/17200
Two adaptive rejection sampling schemes for probability density functions log-convex tails
Martino, Luca; Míguez Arenas, Joaquín
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.
Documento depositado en el repositorio arXiv.org. Versión: arXiv:1111.4942v1 [stat.CO]
2011-11-21T00:00:00ZOn the Generalized Ratio of Uniforms as a Combination of Transformed Rejection and Extended Inverse of Density Sampling
http://hdl.handle.net/10016/17175
On the Generalized Ratio of Uniforms as a Combination of Transformed Rejection and Extended Inverse of Density Sampling
Martino, Luca; Luengo García, David; Míguez Arenas, Joaquín
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.
Documento depositado en el repositorio arXiv.org. Versión: arXiv:1205.0482v6 [stat.CO]
2012-08-11T00:00:00ZEfficient random variable generation: ratio of uniforms and polar rejection sampling
http://hdl.handle.net/10016/16645
Efficient random variable generation: ratio of uniforms and polar rejection sampling
Luengo García, David; Martino, Luca
Monte Carlo techniques, which require the generation of samples from some target density, are often the only alternative for performing Bayesian inference. Two classic sampling techniques to draw independent samples are the ratio of uniforms (RoU) and rejection sampling (RS). An efficient sampling algorithm is proposed combining the RoU and polar RS (i.e. RS inside a sector of a circle using polar coordinates). Its efficiency is shown in drawing samples from truncated Cauchy and Gaussian random variables, which have many important applications in signal processing and communications.
2012-03-01T00:00:00ZEfficient sampling from truncated bivariate Gaussians via Box-Muller transformation
http://hdl.handle.net/10016/16644
Efficient sampling from truncated bivariate Gaussians via Box-Muller transformation
Martino, Luca; Luengo García, David; Míguez Arenas, Joaquín
Many practical simulation tasks demand procedures to draw samples efficiently from multivariate truncated Gaussian distributions. Introduced is a novel rejection approach, based on the Box-Muller transformation, to generate samples from a truncated bivariate Gaussian density with an arbitrary support. Furthermore, for an important class of support regions the new method allows exact sampling to be achieved, thus becoming the most efficient approach possible.Many practical simulation tasks demand procedures to draw samples efficiently from multivariate truncated Gaussian distributions. Introduced is a novel rejection approach, based on the Box-Muller transformation, to generate samples from a truncated bivariate Gaussian density with an arbitrary support. Furthermore, for an important class of support regions the new method allows exact sampling to be achieved, thus becoming the most efficient approach possible.
2012-11-01T00:00:00Z