Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages. (English) Zbl 1078.62106

James, Lancelot F.

Ann. Stat. 33, No. 4, 1771-1799 (2005).

Summary: This article develops, and describes how to use, results concerning disintegrations of Poisson random measures. These results are fashioned as simple tools that can be tailor-made to address inferential questions arising in a wide range of Bayesian nonparametric and spatial statistical models. The Poisson disintegration method is based on the formal statement of two results concerning a Laplace functional change of measure and a Poisson Palm/Fubini calculus in terms of random partitions of the integers \(\{1,\dots,n\}\). The techniques are analogous to, but much more general than, techniques for the Dirichlet process and weighted gamma process developed by A. Y. Lo [Ann. Stat. 12, 351–357 (1984; Zbl 0557.62036)] and by A. Y. Lo and C.-S. Weng [Ann. Inst. Stat. Math. 41, 227–245 (1989; Zbl 0716.62043)]. In order to illustrate the flexibility of the approach, large classes of random probabilities measures and random hazards or intensities which can be expressed as functionals of Poisson random measures are described. We describe a unified posterior analysis of classes of discrete random probabilities which identifies and exploits features common to all these models.
The analysis circumvents many of the difficult issues involved in Bayesian nonparametric calculus, including a combinatorial component. This allows one to focus on the unique features of each process which are characterized via real valued functions \(h\). The applicability of the technique is further illustrated by obtaining explicit posterior expressions for Lévy-Cox moving average processes within the general setting of multiplicative intensity models. In addition, novel computational procedures, similar to efficient procedures developed for the Dirichlet process, are briefly discussed for these models.

MSC:

62M99	Inference from stochastic processes
60G57	Random measures
62G99	Nonparametric inference
62M30	Inference from spatial processes
62G05	Nonparametric estimation
62F15	Bayesian inference

Keywords:

cumulants; inhomogeneous Poisson process; Lévy measure; generalized gamma process; weighted Chinese restaurant; multiplicative intensity models

Citations:

Zbl 0557.62036; Zbl 0716.62043

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