Stein's method of exchangeable pairs for absolutely continuous, univariate distributions with applications to the Polya urn model

Döbler, Christian

doi:10.48550/arxiv.1207.0533

Cited by 9 publications

(30 citation statements)

References 11 publications

(23 reference statements)

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“…Taking differences then allows us to estimate the expectation of the right hand side of (6) when w is replaced by W n by exploiting the similarity of the two characterizing operators; a similar argument can be found in [14] and [18] for stationary distributions of birth-death chains. The results most closely related to the present work is [11], and its connections to the present manuscript are discussed in Remark 3.2 First we introduce some notation. We say a subset I of the integers Z is a finite integer…”

Section: Characterizing Equations and Generatorsmentioning

confidence: 74%

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Stein's Method for the Beta Distribution and the Pólya-Eggenberger Urn

Goldstein¹,

Reinert²

2013

J. Appl. Probab.

116

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Using a characterizing equation for the beta distribution, Stein's method is applied to obtain bounds of the optimal order for the Wasserstein distance between the distribution of the scaled number of white balls drawn from a Pólya-Eggenberger urn and its limiting beta distribution. The bound is computed by making a direct comparison between characterizing operators of the target and the beta distribution, the former derived by extending Stein's density approach to discrete distributions. In addition, refinements are given to Döbler's (2012) result for the arcsine approximation for the fraction of time a simple random walk of even length spends positive, and so also to the distributions of its last return time to 0 and its first visit to its terminal point, by supplying explicit constants to the present Wasserstein bound and also demonstrating that its rate is of the optimal order.

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Section: Characterizing Equations and Generatorsmentioning

confidence: 74%

“…Let x ∧ y and x ∨ y denote the minimum and maximum of two real numbers x and y, respectively. Connections between Theorem 1.1 and the work [11] of Döbler are spelled out in Remark 3.2.…”

Section: Introductionmentioning

confidence: 92%

Stein's Method for the Beta Distribution and the Pólya-Eggenberger Urn

Goldstein¹,

Reinert²

2013

J. Appl. Probab.

116

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“…In this section we first prove the variant of Corollary 2.16 of [5] which we use in our paper. It includes Lemma 2.7 as a special case.…”

Section: Technical Resultsmentioning

confidence: 99%

“…Proof. We prove the three items separately, closely following [5] and in particular his Lemma 5. To deal with this last expression we use the identities We conclude the paper with a proof of Proposition 3.2, restated for convenience.…”

Section: Technical Resultsmentioning

confidence: 99%

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

Ley¹,

Reinert²,

Swan³

2015

Preprint

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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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“…We do not recover their results exactly, because in that paper the equations are extended to the real line. See also [14] (i.e. the arXiv version of [15]) for an in depth first study of the problem of extending Stein equations outside the support of the target.…”

Section: Stein Factorsmentioning

confidence: 99%

Distances between distributions via Stein's method

Ernst¹,

Swan²

2019

Preprint

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We build on the formalism developed in [21] to propose new representations of solutions to Stein equations. We provide new uniform and non uniform bounds on these solutions (a.k.a. Stein factors). We use these representations to obtain representations for differences between expectations in terms of solutions to the Stein equations. We apply these to compute abstract Stein-type bounds on Kolmogorov, Total Variation and Wasserstein distances between arbitrary distributions. We apply our results to several illustrative examples, and compare our results with current literature on the same topic, whenever possible. In all occurrences our results are competitive.

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Stein's method of exchangeable pairs for absolutely continuous, univariate distributions with applications to the Polya urn model

Cited by 9 publications

References 11 publications

Stein's Method for the Beta Distribution and the Pólya-Eggenberger Urn

Stein's Method for the Beta Distribution and the Pólya-Eggenberger Urn

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

Distances between distributions via Stein's method

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