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Poisson process partition calculus with an application to Bayesian . . .
, 2005
"... 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 tailormade to address inferential questions arising in a wide range of Bayesian nonparametric and spatial statistical models. The P ..."
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Cited by 42 (10 self)
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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 tailormade 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
Regenerative composition structures
 ANN. PROBAB
, 2005
"... A new class of random composition structures (the ordered analog of Kingman’s partition structures) is defined by a regenerative description of component sizes. Each regenerative composition structure is represented by a process of random sampling of points from an exponential distribution on the po ..."
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Cited by 31 (19 self)
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A new class of random composition structures (the ordered analog of Kingman’s partition structures) is defined by a regenerative description of component sizes. Each regenerative composition structure is represented by a process of random sampling of points from an exponential distribution on the positive halfline, and separating the points into clusters by an independent regenerative random set. Examples are composition structures derived from residual allocation models, including one associated with the Ewens sampling formula, and composition structures derived from the zero set of a Brownian motion or Bessel process. We provide characterisation results and formulas relating the distribution of the regenerative composition to the Lévy parameters of a subordinator whose range is the corresponding regenerative set. In particular, the only reversible regenerative composition structures are those associated with the interval partition of [0, 1] generated by excursions of a standard Bessel bridge of dimension 2 − 2α for some α ∈ [0, 1].
Regeneration in Random Combinatorial Structures
, 2009
"... Theory of Kingman’s partition structures has two culminating points • the general paintbox representation, relating finite partitions to hypothetical infinite populations via a natural sampling procedure, • a central example of the theory: the EwensPitman twoparameter partitions. In these notes we ..."
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Cited by 2 (2 self)
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Theory of Kingman’s partition structures has two culminating points • the general paintbox representation, relating finite partitions to hypothetical infinite populations via a natural sampling procedure, • a central example of the theory: the EwensPitman twoparameter partitions. In these notes we further develop the theory by • passing to structures enriched by the order on the collection of categories, • extending the class of tractable models by exploring the idea of regeneration, • analysing regenerative properties of the EwensPitman partitions, • studying asymptotic features of the regenerative compositions.
Regenerative tree growth: Markovian embedding of fragmenters, bifurcators and bead splitting processes ∗
, 2013
"... Some, but not all processes of the form Mt = exp(−ξt) for a purejump subordinator ξ with Laplace exponent Φ arise as residual mass processes of particle 1 (tagged particle) in an exchangeable fragmentation processes. We introduce the notion of a Markovian embedding of M in an exchangeable fragmenta ..."
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Cited by 1 (1 self)
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Some, but not all processes of the form Mt = exp(−ξt) for a purejump subordinator ξ with Laplace exponent Φ arise as residual mass processes of particle 1 (tagged particle) in an exchangeable fragmentation processes. We introduce the notion of a Markovian embedding of M in an exchangeable fragmentation process and show that for each Φ, there is a unique binary dislocation measure ν such that M has a Markovian embedding in an associated exchangeable fragmentation process. The identification of the Laplace exponent Φ ∗ of its tagged particle process M ∗ gives rise to a symmetrisation operation Φ ↦ → Φ ∗ , which we investigate in a general study of pairs (M,M ∗ ) that coincide up to a junction time and then evolve independently. We call M a fragmenter and (M,M ∗ ) a bifurcator. For all Φ and α> 0, we can represent a fragmenter M as an interval R1 = [0, ∫ ∞ 0 Mα t dt] equipped with a purely atomic probability measure µ1 capturing the jump sizes of Mt after an αselfsimilar timechange. We call (R1,µ1) an (α,Φ)string of beads. We study binary tree growth processes that in the nth step sample a bead from µn and build (Rn+1,µn+1) by splitting the bead into a new string of beads, a rescaled independent copy of (R1,µ1) that we tie to the position of the sampled bead. We show that all such bead splitting processes converge almost surely to an αselfsimilar CRT, in the GromovHausdorffProhorov sense.
Spatial Neutral to the Right Species Sampling Mixture Models
, 2006
"... This paper describes briefly how one may utilize a class of species sampling mixture models derived from Doksum’s (1974) neutral to the right processes. For practical implementation we describe an ordered/ranked variant of the generalized weighted Chinese restaurant process. 1 ..."
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This paper describes briefly how one may utilize a class of species sampling mixture models derived from Doksum’s (1974) neutral to the right processes. For practical implementation we describe an ordered/ranked variant of the generalized weighted Chinese restaurant process. 1