Random Variable Generation Using Concavity Properties of Transformed Densities

Evans, Michael; Swartz, Tim B.

doi:10.1080/10618600.1998.10474792

Cited by 31 publications

(45 citation statements)

References 7 publications

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“…Evans and Swartz proportional squeeze Slightly different is the situation for the variant of TDR suggested by Evans and Swartz (3). To obtain a simpler sampling algorithm they suggest to take π i = p i and π i+1 = p i+1 .…”

Section: Original Variant Midpoint Methodsmentioning

confidence: 99%

Asymptotically Optimal Design Points for Rejection Algorithms

Derflinger

Hörmann

2005

Communications in Statistics - Simulation and Computation

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Very fast automatic rejection algorithms were developed recently which allow to generate random variates from large classes of unimodal distributions. They require the choice of several design points which decompose the domain of the distribution into small sub-intervals.The optimal choice of these points is an important but unsolved problem. So we present an approach that allows to characterize optimal design points in the asymptotic case (when their number tends to infinity) under mild regularity conditions. We describe a short algorithm to calculate these asymptotically optimal points in practice. Numerical experiments indicate that they are very close to optimal even when only six or seven design points are calculated.

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Section: Original Variant Midpoint Methodsmentioning

confidence: 99%

Asymptotically Optimal Design Points for Rejection Algorithms

Derflinger

Hörmann

2005

Communications in Statistics - Simulation and Computation

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“…Moreover, one of the design points must be the pole itself. This is the approach suggested by Evans and Swartz [1998] for the F-distribution when its density is unbounded.…”

Section: Transformed Density Rejection and Polesmentioning

confidence: 99%

Inverse transformed density rejection for unbounded monotone densities

Hörmann

Leydold

Derflinger

2007

ACM Trans. Model. Comput. Simul.

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A new algorithm for sampling from largely arbitrary monotone, unbounded densities is presented. The user has to provide a program to evaluate the density and its derivative and the location of the pole. Then the setup of the new algorithm constructs different hat functions for the pole region and for the tail region, respectively. For the pole region a new method is developed that uses a transformed density rejection hat function of the inverse density. As the order of the pole is calculated in the setup, conditions that guarantee the correctness of the constructed hat functions are provided. Numerical experiments indicate that the new algorithm works correctly and moderately fast for many different unbounded densities.

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“…Assume that, Y = y = [2,5] . The simple estimates are X = {x 1 = log(2), x 2 = − log(5)}, and, therefore, we can restrict the search of the bound to the interval I = [min(X ) = − log(5), max(X ) = log(2)] (note that we omit the subscript because we have just one set, B 1 ≡ R).…”

Section: A Example 1: Calculation Of Upper Bounds For the Likelihoodmentioning

confidence: 99%

“…Unfortunately, this procedure is only valid when the target pdf is strictly log-concave, which is not the case in most practical cases. Although an extension has been proposed [9,5] that enables the application of the ARS algorithm with T -concave distributions (where T is a monotonically increasing function, not necessarily the logarithm), it does not address the main limitations of the original method (e.g., the impossibility to draw from multimodal distributions) and is hard to apply, due to the difficulty to find adequate T transformations other than the logarithm. Another algorithm, called adaptive rejection metropolis sampling (ARMS) [14], is an attempt to extend the ARS to multimodal densities by adding Metropolis-Hastings steps.…”

Section: Introductionmentioning

confidence: 99%

Generalized rejection sampling schemes and applications in signal processing

Martino

Mı́guez

2010

Signal Processing

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Bayesian methods and their implementations by means of sophisticated Monte Carlo techniques, such as Markov chain Monte Carlo (MCMC) and particle filters, have become very popular in signal processing over the last years. However, in many problems of practical interest these techniques demand procedures for sampling from probability distributions with non-standard forms, hence we are often brought back to the consideration of fundamental simulation algorithms, such as rejection sampling (RS). Unfortunately, the use of RS techniques demands the calculation of tight upper bounds for the ratio of the target probability density function (pdf) over the proposal density from which candidate samples are drawn. Except for the class of log-concave target pdf's, for which an efficient algorithm exists, there are no general methods to analytically determine this bound, which has to be derived from scratch for each specific case. In this paper, we introduce new schemes for (a) obtaining upper bounds for likelihood functions and (b) adaptively computing proposal densities that approximate the target pdf closely. The former class of methods provides the tools to easily sample from a posteriori probability distributions (that appear very often in signal processing problems) by drawing candidates from the prior distribution. However, they are even more useful when they are exploited to derive the generalized adaptive RS (GARS) algorithm introduced in the second part of the paper. The proposed GARS method yields a sequence of proposal densities that converge towards the target pdf and enable a very efficient sampling of a broad class of probability distributions, possibly with multiple modes and non-standard forms. We provide some simple numerical examples to illustrate the use of the proposed techniques, including an example of target localization using range measurements, often encountered in sensor network applications.

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Random Variable Generation Using Concavity Properties of Transformed Densities

Cited by 31 publications

References 7 publications

Asymptotically Optimal Design Points for Rejection Algorithms

Asymptotically Optimal Design Points for Rejection Algorithms

Inverse transformed density rejection for unbounded monotone densities

Generalized rejection sampling schemes and applications in signal processing

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