Bayesian Analysis

Stochastic Approximations to the Pitman–Yor Process

Julyan Arbel, Pierpaolo De Blasi, and Igor Prünster

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In this paper we consider approximations to the popular Pitman–Yor process obtained by truncating the stick-breaking representation. The truncation is determined by a random stopping rule that achieves an almost sure control on the approximation error in total variation distance. We derive the asymptotic distribution of the random truncation point as the approximation error ϵ goes to zero in terms of a polynomially tilted positive stable random variable. The practical usefulness and effectiveness of this theoretical result is demonstrated by devising a sampling algorithm to approximate functionals of the ϵ-version of the Pitman–Yor process.

Article information

Bayesian Anal., Volume 14, Number 4 (2019), 1201-1219.

First available in Project Euclid: 11 June 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E20: Asymptotic distribution theory
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures

stochastic approximation asymptotic distribution Bayesian Nonparametrics Pitman–Yor process random functionals random probability measure stopping rule

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Arbel, Julyan; De Blasi, Pierpaolo; Prünster, Igor. Stochastic Approximations to the Pitman–Yor Process. Bayesian Anal. 14 (2019), no. 4, 1201--1219. doi:10.1214/18-BA1127.

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Supplemental materials

  • Supplementary Material of “Stochastic Approximations to the Pitman–Yor Process”. Algorithm 3 for generating from a polynomially tilted positive stable random variable (in a separate document).