Poisson Process Partition Calculus with applications to Exchangeable models and Bayesian Nonparametrics

By Lancelot F. James. 2002

128 citations recorded by the source corpus.

This article discusses the usage of a partiton based Fubini calculus for Poisson processes. The approach is an amplification of Bayesian techniques developed in Lo and Weng for gamma/Dirichlet processes. Applications to models are considered which all fall within an inhomogeneous spatial extension of the size biased framework used in Perman, Pitman and Yor. Among some of the results; an explicit partition based calculus is then developed for such models, which also includes a series of important exponential change of measure formula. These results are applied to obtain results for Levy-Cox models, identities related to the two-parameter Poisson-Dirichlet process and other processes, generalisations of the Markov-Krein correspondence, calculus for extended Neutral to the Right processes, among other things.

Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8efb434d2ea63ee029db5db480a336e9539bf071

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Citations in the corpus

Bayesian Nonparametrics: Models beyond the Dirichlet process (2009, unknown): “The first one, set forth in James (2002) and developed and refined in subsequent papers, is termed Poisson partition calculus: the key idea consists in facing the problem at the level of the Poisson process underlying the CRM, according to (7), and then to use Fubini– type arguments.”
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 .”
Conjugacy as a Distinctive Feature of the Dirichlet Process (2006, methodology): “On the contrary, if one makes the additional assumption of diffuseness of , it is possible to recover a nice result due to Pitman (2003) (see also Prünster, 2002; James, 2002).”
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 Poisson process partition calculus with an application to Bayesian Lévy moving averages (2005, background): “James [20] shows that Doksum’s [8] neutral to the right processes can be obtained by the choice ofh(s, y,N) = se−Z(y−), for (s, y) in [0,1] × (0,∞), whereZ(y−) = ∫ 10 ∫ ∞0 I{x y}[− log(1 − u)]N(du,dx), where nowρ(ds|y) is a Lévy measure on[0,1] andη is modeled as a cumulative hazard.”
Combinatorial Stochastic Processes (2006, background): “The Chinese Restaurant Process and associated computations with random partitions have found applications in Bayesian statistics [109, 287, 210, 212], and in the theory of representations of the infinite symmetric group [243].”
Bayesian Nonparametric Analysis for a Generalized Dirichlet Process Prior (2005, methodology): “(See also James, 2002; Prünster, 2002) for different derivations of this result.”
Generalized weighted Chinese restaurant processes for species sampling mixture models (2003, unknown): “These were discussed recently in James (2002).”
Poisson-Kingman partitions (2002, methodology): “This paper presents some general formulas for random partitions of a finite set derived by Kingman's model of random sampling from an interval partition generated by subintervals whose lengths are the points of a Poisson point process.”
Posterior Analysis for Normalized Random Measures with Independent Increments (2009, methodology): “…with independent increments (NRMI) is formally introduced as the normalization of suitably time-changed independent increment processes and distributional results for their means derived: this work shows that, at least in terms of means, such processes are indeed tractable (see also James, 2002).”