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August 2005 Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages
Lancelot F. James
Ann. Statist. 33(4): 1771-1799 (August 2005). DOI: 10.1214/009053605000000336


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,…,n}. The techniques are analogous to, but much more general than, techniques for the Dirichlet process and weighted gamma process developed in [Ann. Statist. 12 (1984) 351–357] and [Ann. Inst. Statist. Math. 41 (1989) 227–245]. In order to illustrate the flexibility of the approach, large classes of random probability 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 probability 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.


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Lancelot F. James. "Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages." Ann. Statist. 33 (4) 1771 - 1799, August 2005.


Published: August 2005
First available in Project Euclid: 5 August 2005

zbMATH: 1078.62106
MathSciNet: MR2166562
Digital Object Identifier: 10.1214/009053605000000336

Primary: 62G05
Secondary: 62F15

Keywords: Cumulants , generalized gamma process , inhomogeneous Poisson process , Lévy measure , multiplicative intensity model , weighted Chinese restaurant

Rights: Copyright © 2005 Institute of Mathematical Statistics


Vol.33 • No. 4 • August 2005
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