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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 32 (18 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].
Exchangeable Gibbs partitions and Stirling triangles
"... For two collections of nonnegative and suitably normalised weights W = (Wj) and V = (Vn,k), a probability distribution on the set of partitions of the set {1,...,n} is defined by assigning to a generic partition {Aj, j ≤ k} the probability Vn,k W A1  · · ·W Ak, where Aj  is the number of ele ..."
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Cited by 22 (5 self)
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For two collections of nonnegative and suitably normalised weights W = (Wj) and V = (Vn,k), a probability distribution on the set of partitions of the set {1,...,n} is defined by assigning to a generic partition {Aj, j ≤ k} the probability Vn,k W A1  · · ·W Ak, where Aj  is the number of elements of Aj. We impose constraints on the weights by assuming that the resulting random partitions Πn of [n] are consistent as n varies, meaning that they define an exchangeable partition of the set of all natural numbers. This implies that the weights W must be of a very special form depending on a single parameter α ∈ [−∞, 1]. The case α = 1 is trivial, and for each value of α ̸ = 1 the set of possible Vweights is an infinitedimensional simplex. We identify the extreme points of the simplex by solving the boundary problem for a generalised Stirling triangle. In particular, we show that the boundary is discrete for − ∞ ≤ α < 0 and continuous for 0 ≤ α < 1. For α ≤ 0 the extremes correspond to the members of the EwensPitman family of random partitions indexed by (α, θ), while for 0 < α < 1 the extremes are obtained by conditioning an (α, θ)partition on the asymptotics of the number of blocks of Πn as n tends to infinity.
Poisson calculus for spatial neutral to the right processes
, 2003
"... In this paper we consider classes of nonparametric priors on spaces of distribution functions and cumulative hazard measures that are based on extensions of the neutral to the right (NTR) concept. In particular, spatial neutral to the right processes that extend the NTR concept from priors on the cl ..."
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Cited by 10 (1 self)
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In this paper we consider classes of nonparametric priors on spaces of distribution functions and cumulative hazard measures that are based on extensions of the neutral to the right (NTR) concept. In particular, spatial neutral to the right processes that extend the NTR concept from priors on the class of distributions on the real line to classes of distributions on general spaces are discussed. Representations of the posterior distribution of the spatial NTR processes are given. A different type of calculus than traditionally employed in the Bayesian literature, based on Poisson process partition calculus methods described in James (2002), is provided which offers a streamlined proof of posterior results for NTR models and its spatial extension. The techniques are applied to progressively more complex models ranging from the complete data case to semiparametric multiplicative intensity models. Refinements are then given which describes the underlying properties of spatial NTR processes analogous to those developed for the Dirichlet process. The analysis yields accessible moment formulae and characterizations of the posterior distribution and relevant marginal distributions. An EPPF formula and additionally a distribution related to the risk and death sets is computed. In the homogeneous case, these distributions turn out to be connected and overlap with recent work on regenerative compositions defined by suitable discretisation of subordinators. The formulae we develop for the marginal distribution of spatial NTR models provide clues on how to sample posterior distributions in complex settings. In addition the spatial NTR is further extended to the mixture model setting which allows for applicability of such processes to much more complex data structures. A description of a species sampling model derived from a spatial NTR model is also given.
Characterizations of exchangeable partitions and random discrete distributions by deletion properties
, 2009
"... We prove a longstanding conjecture which characterises the EwensPitman twoparameter family of exchangeable random partitions, plus a short list of limit and exceptional cases, by the following property: for each n = 2,3,..., if one of n individuals is chosen uniformly at random, independently of ..."
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Cited by 3 (2 self)
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We prove a longstanding conjecture which characterises the EwensPitman twoparameter family of exchangeable random partitions, plus a short list of limit and exceptional cases, by the following property: for each n = 2,3,..., if one of n individuals is chosen uniformly at random, independently of the random partition πn of these individuals into various types, and all individuals of the same type as the chosen individual are deleted, then for each r> 0, given that r individuals remain, these individuals are partitioned according to π ′ r for some sequence of random partitions (π ′ r) that does not depend on n or r. An analogous result characterizes the associated PoissonDirichlet family of random discrete distributions by an independence property related to random deletion of a frequency chosen by a sizebiased pick. We also survey the regenerative properties of members of the twoparameter family, and settle a question regarding the explicit arrangement of intervals with lengths given by the terms of the PoissonDirichlet random sequence into the interval partition induced by the range of a neutraltothe right process.
On a Gibbs characterization of normalized generalized Gamma processes
, 707
"... We show that a Gibbs characterization of normalized generalized Gamma processes, recently obtained in Lijoi, Prünster and Walker (2007), can alternatively be derived by exploiting a characterization of exponentially tilted PoissonKingman models stated in Pitman (2003). We also provide a completion ..."
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Cited by 2 (0 self)
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We show that a Gibbs characterization of normalized generalized Gamma processes, recently obtained in Lijoi, Prünster and Walker (2007), can alternatively be derived by exploiting a characterization of exponentially tilted PoissonKingman models stated in Pitman (2003). We also provide a completion of this result investigating the existence of normalized random measures inducing exchangeable Gibbs partitions of type α ∈ (−∞,0].
StickBreaking Autoregressive Processes
"... This paper considers the problem of defining a timedependent nonparametric prior for use in Bayesian nonparametric modelling of time series. A recursive construction allows the definition of priors whose marginals have a general stickbreaking form. The processes with PoissonDirichlet and Dirichle ..."
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Cited by 2 (0 self)
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This paper considers the problem of defining a timedependent nonparametric prior for use in Bayesian nonparametric modelling of time series. A recursive construction allows the definition of priors whose marginals have a general stickbreaking form. The processes with PoissonDirichlet and Dirichlet process marginals are investigated in some detail. We develop a general conditional Markov Chain Monte Carlo (MCMC) method for inference in the wide subclass of these models where the parameters of the marginal stickbreaking process are nondecreasing sequences. We derive a generalized Pólya urn scheme type representation of the Dirichlet process construction, which allows us to develop a marginal MCMC method for this case. We apply the proposed methods to financial data to develop a semiparametric stochastic volatility model with a timevarying nonparametric returns distribution. Finally, we present two examples concerning the analysis of regional GDP and its growth.
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.
Generalized Chinese restaurant construction of exchangeable Gibbs partitions and related results. ∗†
, 805
"... By resorting to sequential constructions of exchangeable random partitions (Pitman, 2006), and exploiting some known facts about generalized Stirling numbers, we derive a generalized Chinese restaurant process construction of exchangeable Gibbs partitions of type α (Gnedin and Pitman, 2006). Our con ..."
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By resorting to sequential constructions of exchangeable random partitions (Pitman, 2006), and exploiting some known facts about generalized Stirling numbers, we derive a generalized Chinese restaurant process construction of exchangeable Gibbs partitions of type α (Gnedin and Pitman, 2006). Our construction represents the natural theoretical probabilistic framework in which to embed some recent results about a Bayesian nonparametric treatment of estimation problems arising in genetic experiment under Gibbs, species sampling, models priors.
Bayesian density estimation and model selection using nonparametric
, 2008
"... hierarchical mixtures ..."
conditional Gibbs structures
"... WORKING PAPER SERIESBayesian nonparametric estimators derived from ..."