Use of exchangeable pairs in the analysis of simulations

Stein, C. Michael; Diaconis, Persi; Holmes, Susan; Reinert, Gesine

doi:10.1214/lnms/1196283797

Cited by 92 publications

(81 citation statements)

References 22 publications

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“…Such an assumption is in general too strong (see e.g. [28] for a discussion about the arcsine distribution) and weaker assumptions on p are permitted in our framework, although in such cases stronger constraints on the functions in F (P ) are necessary. In particular the constant functions may not belong to F (P ).…”

Section: Notations and Definitionsmentioning

confidence: 99%

“…The central aim of this paper is to provide meaningful bounds on d W (P 1 , P 2 ) in terms of π 0 . Our approach to this problem relies on Stein's density approach introduced in [27,28], as further developed in [15,16,17,18]. Let P 1 have density p 1 with interval support I 1 with closure [a 1 , b 1 ] for some −∞ ≤ a 1 < b 1 ≤ +∞.…”

Section: Introductionmentioning

confidence: 99%

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Distances between nested densities and a measure of the impact of the prior in Bayesian statistics

Ley¹,

Reinert²,

Swan³

2017

Ann. Appl. Probab.

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In this paper we propose tight upper and lower bounds for the Wasserstein distance between any two univariate continuous distributions with probability densities p 1 and p 2 having nested supports. These explicit bounds are expressed in terms of the derivative of the likelihood ratio p 1 /p 2 as well as the Stein kernel τ 1 of p 1 . The method of proof relies on a new variant of Stein's method which manipulates Stein operators.We give several applications of these bounds. Our main application is in Bayesian statistics : we derive explicit data-driven bounds on the Wasserstein distance between the posterior distribution based on a given prior and the no-prior posterior based uniquely on the sampling distribution. This is the first finite sample result confirming the well-known fact that with well-identified parameters and large sample sizes, reasonable choices of prior distributions will have only minor effects on posterior inferences if the data are benign.

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Section: Notations and Definitionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Distances between nested densities and a measure of the impact of the prior in Bayesian statistics

Ley¹,

Reinert²,

Swan³

2017

Ann. Appl. Probab.

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“…Characterizations like (1.1) have been established en mass [for an overview on characterizing Stein operators and further references, we recommend the work by Ley et al (2017)]. Charles Stein himself presented some ideas fundamental to the so-called density approach [see Stein (1986), Chapter VI, and Stein et al (2004), Section 5] which we shall use as the basis of our considerations. Related results for the special case of exponential families were already given by Hudson (1978) and Prakasa Rao (1979).…”

Section: Introductionmentioning

confidence: 99%

Fixed point characterizations of continuous univariate probability distributions and their applications

Betsch

Ebner

2019

Ann Inst Stat Math

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By extrapolating the explicit formula of the zero-bias distribution occurring in the context of Stein's method, we construct characterization identities for a large class of absolutely continuous univariate distributions. Instead of trying to derive characterizing distributional transformations that inherit certain structures for the use in further theoretic endeavours, we focus on explicit representations given through a formula for the density-or distribution function. The results we establish with this ambition feature immediate applications in the area of goodness-of-fit testing. We draw up a blueprint for the construction of tests of fit that include procedures for many distributions for which little (if any) practicable tests are known.To illustrate this last point, we construct a test for the Burr Type XII distribution for which, to our knowledge, not a single test is known aside from the classical universal procedures.MSC 2010 subject classifications. Primary 62E10 Secondary 60E10, 62G10Over the last decades, Stein's method for distributional approximation has become a viable tool for proving limit theorems and establishing convergence rates. At it's heart lies the well-known Stein characterization which states that a real-valued random variable Z has a standard normal distribution if, and only if,holds for all functions f of a sufficiently large class of test functions. To exploit this characterization for testing the hypothesisof normality, where P X is the distribution of a real-valued random variable X, against general alternatives, Betsch and Ebner (2019b) used that (1.1) can be untied from the class of test functions with the help of the so-called zero-bias transformation introduced by Goldstein and Reinert (1997). To be specific, a real-valued random variable X * is said to have the X-zero-bias distri-bution ifholds for any of the respective test functions f . If EX = 0 and Var(X) = 1, the X-zero-bias distribution exists and is unique, and it has distribution function(1.3)By (1.1), the standard Gaussian distribution is the unique fixed point of the transformation P X → P X * . Thus, the distribution of X is standard normal if, and only if,where F X denotes the distribution function of X. In the spirit of characterization-based goodnessof-fit tests, an idea introduced by Linnik (1962), this fixed point property directly admits a new class of testing procedures as follows. Letting T X n be an empirical version of T X and F n the empirical distribution function, both based on the standardized sample, Betsch and Ebner (2019b) proposed a test for (1.2) based on the statisticwhere w is an appropriate weight function, which, in view of (1.4), rejects the normality hypothesis for large values of G n . As these tests have several desirable properties such as consistency against general alternatives, and since they show a very promising performance in simulations, we devote this work to the question to what extent the fixed point property and the class of goodness-of-fit procedures may be generalized to othe...

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“…In a different direction, Stein et al. (2004) drew a connection between the use of control variates in MCMC methods and the ‘exchangeable pairs’ construction that is used in Stein's method for distribution approximation.…”

Section: Introductionmentioning

confidence: 99%

Control Variates for Estimation Based on Reversible Markov Chain Monte Carlo Samplers

Δελλαπόρτας

Kontoyiannis

2011

Journal of the Royal Statistical Society Series B: Statistical Methodology

106

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A general methodology is introduced for the construction and effective application of control variates to estimation problems involving data from reversible Markov chain Monte Carlo samplers. We propose the use of a specific class of functions as control variates, and we introduce a new consistent estimator for the values of the coefficients of the optimal linear combination of these functions. For a specific Markov chain Monte Carlo scenario, the form and proposed construction of the control variates is shown to provide an exact solution of the associated Poisson equation. This implies that the estimation variance in this case (in the central limit theorem regime) is exactly zero. The new estimator is derived from a novel, finite dimensional, explicit representation for the optimal coefficients. The resulting variance reduction methodology is primarily (though certainly not exclusively) applicable when the simulated data are generated by a random-scan Gibbs sampler. Markov chain Monte Carlo examples of Bayesian inference problems demonstrate that the corresponding reduction in the estimation variance is significant, and that in some cases it can be quite dramatic. Extensions of this methodology are discussed and simulation examples are presented illustrating the utility of the methods proposed. All methodological and asymptotic arguments are rigorously justified under essentially minimal conditions.

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Use of exchangeable pairs in the analysis of simulations

Cited by 92 publications

References 22 publications

Distances between nested densities and a measure of the impact of the prior in Bayesian statistics

Distances between nested densities and a measure of the impact of the prior in Bayesian statistics

Fixed point characterizations of continuous univariate probability distributions and their applications

Control Variates for Estimation Based on Reversible Markov Chain Monte Carlo Samplers

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