Bayesian Analysis

Stochastic Approximations to the Pitman–Yor Process

Julyan Arbel, Pierpaolo De Blasi, and Igor Prünster

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Abstract

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

Source
Bayesian Anal., Advance publication (2018), 19 pages.

Dates
First available in Project Euclid: 11 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.ba/1560240026

Digital Object Identifier
doi:10.1214/18-BA1127

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

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

Rights
Creative Commons Attribution 4.0 International License.

Citation

Arbel, Julyan; De Blasi, Pierpaolo; Prünster, Igor. Stochastic Approximations to the Pitman–Yor Process. Bayesian Anal., advance publication, 11 June 2019. doi:10.1214/18-BA1127. https://projecteuclid.org/euclid.ba/1560240026


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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).