Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages

By Lancelot F. James. 2005

112 citations recorded by the source corpus.

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) 351-3571 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 Levy-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.

Subfields: Applied Mathematics, Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/28da3b68ad7f82b35e4c2d996ea8f3b3a39b217c

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Citations in the corpus

Bayesian inference with dependent normalized completely random measures (2014, methodology): “See [6,20] and, for example, [17] for uses of representation (3) for Bayesian modeling.”
Posterior Analysis for Normalized Random Measures with Independent Increments (2009, methodology): “(21) This in turn determines the posterior distribution of ̃ |X. Additionally, given the form of n, statements (ii) and (iii) of theorem 3.2 in James (2005a) can be exploited in order to provide a preliminary description of the posterior distribution of P̃.”
Applied Mathematical Sciences (2008, unknown): “As in Brix [2] and James [10], [11], we define a completely random measure μ(dt) on T as follows:”
Bayesian Nonparametrics: Models beyond the Dirichlet process (2009, unknown): “Given the analytical difficulties we have just sketched, performing a full Bayesian analysis asks for the application of suitable computational schemes such as the MCMC algorithm devised in Petrone (1999b).”
Pricing Options Under a Generalized Markov-Modulated Jump-Diffusion Model (2007, background): “Let dz t denote a Lévy measure on the space × depending on t [19, 20], is a -finite (nonatomic) measure on .”