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Functionals of Dirichlet processes, the Cifarelli-Regazzini identity and beta-gamma processes. (English) Zbl 1071.62026
Summary: Suppose that \(P_\theta(g)\) is a linear functional of a Dirichlet process with shape \(\theta H\), where \(\theta>0\) 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 D. M. Cifarelli and E. Regazzini [ibid. 18, No. 1, 429–442 (1990; Zbl 0706.62012)] 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.

MSC:
62F15 Bayesian inference
62M99 Inference from stochastic processes
62G99 Nonparametric inference
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