Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Pólya tree posterior distributions on densities

Ismaël Castillo

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Abstract

Pólya trees form a popular class of prior distributions used in Bayesian nonparametrics. For some choice of parameters, Pólya trees are prior distributions on density functions. In this paper we carry out a frequentist analysis of the induced posterior distributions in the density estimation model. We investigate the contraction rate of Pólya tree posterior densities in terms of the supremum loss and study the limiting shape distribution. A nonparametric Bernstein–von Mises theorem is established, as well as a Bayesian Donsker theorem for the posterior cumulative distribution function.

Résumé

Les arbres de Pólya constituent une classe de lois a priori très utilisée en bayésien non-paramétrique. Pour certains choix de paramètres, les arbres de Pólya induisent des lois à densité. Nous menons une analyse fréquentiste des lois a posteriori bayésiennes correspondantes dans le modèle d’estimation de densité. La concentration a posteriori des densités–arbre de Pólya est étudiée en terme de la norme–sup et nous déterminons la loi a posteriori limite après renormalisation. Un théorème de Bernstein–von Mises non-paramétrique est établi, ainsi qu’un théorème de Donsker bayésien pour la fonction de répartition a posteriori.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 2074-2102.

Dates
Received: 23 July 2015
Revised: 8 July 2016
Accepted: 8 August 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1511773739

Digital Object Identifier
doi:10.1214/16-AIHP784

Mathematical Reviews number (MathSciNet)
MR3729648

Zentralblatt MATH identifier
1384.62156

Subjects
Primary: 62G20: Asymptotic properties
Secondary: 62G07: Density estimation 62G15: Tolerance and confidence regions

Keywords
Bayesian nonparametrics Pólya tree distribution Supremum norm convergence Minimax rate Bernstein–von Mises theorem Bayesian Donsker theorem

Citation

Castillo, Ismaël. Pólya tree posterior distributions on densities. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 2074--2102. doi:10.1214/16-AIHP784. https://projecteuclid.org/euclid.aihp/1511773739


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