The Annals of Statistics

Functionals of Dirichlet processes, the Cifarelli–Regazzini identity and Beta-Gamma processes

Lancelot F. James

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Abstract

Suppose that Pθ(g) is a linear functional of a Dirichlet process with shape θH, where θ>0 is the total mass and H is a fixed probability measure. This paper describes how one can use the well-known Bayesian prior to posterior analysis of the Dirichlet process, and a posterior calculus for Gamma processes to ascertain properties of linear functionals of Dirichlet processes. In particular, in conjunction with a Gamma identity, we show easily that a generalized Cauchy–Stieltjes transform of a linear functional of a Dirichlet process is equivalent to the Laplace functional of a class of, what we define as, Beta-Gamma processes. This represents a generalization of an identity due to Cifarelli and Regazzini, which is also known as the Markov–Krein identity for mean functionals of Dirichlet processes. These results also provide new explanations and interpretations of results in the literature. The identities are analogues to quite useful identities for Beta and Gamma random variables. We give a result which can be used to ascertain specifications on H such that the Dirichlet functional is Beta distributed. This avoids the need for an inversion formula for these cases and points to the special nature of the Dirichlet process, and indeed the functional Beta-Gamma calculus developed in this paper.

Article information

Source
Ann. Statist., Volume 33, Number 2 (2005), 647-660.

Dates
First available in Project Euclid: 26 May 2005

Permanent link to this document
https://projecteuclid.org/euclid.aos/1117114332

Digital Object Identifier
doi:10.1214/009053604000001237

Mathematical Reviews number (MathSciNet)
MR2163155

Zentralblatt MATH identifier
1071.62026

Subjects
Primary: 62G05: Estimation
Secondary: 62F15: Bayesian inference

Keywords
Beta-Gamma processes Dirichlet process Markov–Krein identity Gamma process

Citation

James, Lancelot F. Functionals of Dirichlet processes, the Cifarelli–Regazzini identity and Beta-Gamma processes. Ann. Statist. 33 (2005), no. 2, 647--660. doi:10.1214/009053604000001237. https://projecteuclid.org/euclid.aos/1117114332


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