An Efficient Scheme for Sampling in Constrained Domains

Chaudhry, Sharang; Lautzenheiser, Daniel; Ghosh, Kaushik

doi:10.48550/arxiv.2110.10840

“…39/49 and additional relevant sampling methods have been introduced since that review (e.g. [26][27][28]). These sampling methods by themselves can be used to estimate the mean and covariance of ST-MNDs.…”

Section: Discussionmentioning

confidence: 99%

Integral, mean and covariance of the simplex-truncated multivariate normal distribution

Adams¹

2022

Preprint

0

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Compositional data, which is data consisting of fractions or probabilities, is common in many fields including ecology, economics, physical science and political science. If these data would otherwise be normally distributed, their spread can be conveniently represented by a multivariate normal distribution truncated to the non-negative space under a unit simplex. Here this distribution is called the simplex-truncated multivariate normal distribution. For calculations on truncated distributions, it is often useful to obtain rapid estimates of their integral, mean and covariance; these quantities characterising the truncated distribution will generally possess different values to the corresponding non-truncated distribution. In this paper, three different approaches that can estimate the integral, mean and covariance of any simplex-truncated multivariate normal distribution are described and compared. These three approaches are (1) naive rejection sampling, (2) a method described by Gessner et al. that unifies subset simulation and the Holmes-Diaconis-Ross algorithm with an analytical version of elliptical slice sampling, and (3) a semi-analytical method that expresses the integral, mean and covariance in terms of integrals of hyperrectangularly-truncated multivariate normal distributions, the latter of which are readily computed in modern mathematical and statistical packages. Strong agreement is demonstrated between all three approaches, but the most computationally efficient approach depends strongly both on implementation details and the dimension of the simplex-truncated multivariate normal distribution. For computations in low-dimensional distributions, the semi-analytical method is fast and thus should be considered. As the dimension increases, the Gessner et al. method becomes the only practically efficient approach of the methods tested here.

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“…Several methods to efficiently sample from the simplex are available, as reviewed in [6], and additional relevant sampling methods have been introduced since that review (e.g. [26][27][28]). For example, Cong et al [27] provides an algorithm which can be applied to ensure that samples proposed by naive rejection sampling definitely have all fractions (including the redundant fraction) sum to one.…”

Section: Alternative Methods For Calculations On the Simplex-truncate...mentioning

confidence: 99%

Integral, mean and covariance of the simplex-truncated multivariate normal distribution

Adams

¹

2022

PLoS ONE

1

0

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Compositional data, which is data consisting of fractions or probabilities, is common in many fields including ecology, economics, physical science and political science. If these data would otherwise be normally distributed, their spread can be conveniently represented by a multivariate normal distribution truncated to the non-negative space under a unit simplex. Here this distribution is called the simplex-truncated multivariate normal distribution. For calculations on truncated distributions, it is often useful to obtain rapid estimates of their integral, mean and covariance; these quantities characterising the truncated distribution will generally possess different values to the corresponding non-truncated distribution. In this paper, three different approaches that can estimate the integral, mean and covariance of any simplex-truncated multivariate normal distribution are described and compared. These three approaches are (1) naive rejection sampling, (2) a method described by Gessner et al. that unifies subset simulation and the Holmes-Diaconis-Ross algorithm with an analytical version of elliptical slice sampling, and (3) a semi-analytical method that expresses the integral, mean and covariance in terms of integrals of hyperrectangularly-truncated multivariate normal distributions, the latter of which are readily computed in modern mathematical and statistical packages. Strong agreement is demonstrated between all three approaches, but the most computationally efficient approach depends strongly both on implementation details and the dimension of the simplex-truncated multivariate normal distribution. For computations in low-dimensional distributions, the semi-analytical method is fast and thus should be considered. As the dimension increases, the Gessner et al. method becomes the only practically efficient approach of the methods tested here.

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“…Spherical HMC can be viewed as a more efficient special case of Brubaker et al (2012) and Kook et al (2022). A related work SPInS (Chaudhry et al, 2021) maps a hyper-ball containing the constrained domain inside out so that sampler can be defined on a larger unbounded space.…”

Section: Introductionmentioning

confidence: 99%

Sampling constrained continuous probability distributions: A review

Lan

¹

,

Kang

²

2023

WIREs Computational Stats

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The problem of sampling constrained continuous distributions has frequently appeared in many machine/statistical learning models. Many Markov Chain Monte Carlo (MCMC) sampling methods have been adapted to handle different types of constraints on random variables. Among these methods, Hamilton Monte Carlo (HMC) and the related approaches have shown significant advantages in terms of computational efficiency compared with other counterparts. In this article, we first review HMC and some extended sampling methods, and then we concretely explain three constrained HMC‐based sampling methods, reflection, reformulation, and spherical HMC. For illustration, we apply these methods to solve three well‐known constrained sampling problems, truncated multivariate normal distributions, Bayesian regularized regression, and nonparametric density estimation. In this review, we also connect constrained sampling with another similar problem in the statistical design of experiments with constrained design space.This article is categorized under: Applications of Computational Statistics > Computational Mathematics Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory Statistical and Graphical Methods of Data Analysis > Markov Chain Monte Carlo (MCMC) Statistical and Graphical Methods of Data Analysis > Sampling

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An Efficient Scheme for Sampling in Constrained Domains

Cited by 4 publications

References 12 publications

Integral, mean and covariance of the simplex-truncated multivariate normal distribution

Integral, mean and covariance of the simplex-truncated multivariate normal distribution

Integral, mean and covariance of the simplex-truncated multivariate normal distribution

Sampling constrained continuous probability distributions: A review

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