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10
The twoparameter PoissonDirichlet distribution derived from a stable subordinator.
, 1995
"... The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov ..."
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Cited by 221 (37 self)
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The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov chain description due to VershikShmidtIgnatov, are generalized to the twoparameter case. The sizebiased random permutation of pd(ff; `) is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For 0 ! ff ! 1, pd(ff; 0) is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index ff. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950's and 60's. The distribution of ranked lengths of e...
Gibbs Sampling Methods for StickBreaking Priors
"... ... In this paper we present two general types of Gibbs samplers that can be used to fit posteriors of Bayesian hierarchical models based on stickbreaking priors. The first type of Gibbs sampler, referred to as a Polya urn Gibbs sampler, is a generalized version of a widely used Gibbs sampling meth ..."
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Cited by 213 (17 self)
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... In this paper we present two general types of Gibbs samplers that can be used to fit posteriors of Bayesian hierarchical models based on stickbreaking priors. The first type of Gibbs sampler, referred to as a Polya urn Gibbs sampler, is a generalized version of a widely used Gibbs sampling method currently employed for Dirichlet process computing. This method applies to stickbreaking priors with a known P'olya urn characterization; that is priors with an explicit and simple prediction rule. Our second method, the blocked Gibbs sampler, is based on a entirely different approach that works by directly sampling values from the posterior of the random measure. The blocked Gibbs sampler can be viewed as a more general approach as it works without requiring an explicit prediction rule. We find that the blocked Gibbs avoids some of the limitations seen with the Polya urn approach and should be simpler for nonexperts to use.
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].
PoissonKingman Partitions
 of Lecture NotesMonograph Series
, 2002
"... 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. These lengths can be also interpreted as the jumps of a subordin ..."
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Cited by 11 (3 self)
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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. These lengths can be also interpreted as the jumps of a subordinator, that is an increasing process with stationary independent increments. Examples include the twoparameter family of PoissonDirichlet models derived from the Poisson process of jumps of a stable subordinator. Applications are made to the random partition generated by the lengths of excursions of a Brownian motion or Brownian bridge conditioned on its local time at zero.
Gibbs Fragmentation Trees
, 2008
"... We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbstype fragmentation tree with Aldous’ betasplitting model, which has an extended parameter range β>−2 with respect to the beta(β + 1,β + 1) probability distributions on which it is based. In the multif ..."
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Cited by 10 (6 self)
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We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbstype fragmentation tree with Aldous’ betasplitting model, which has an extended parameter range β>−2 with respect to the beta(β + 1,β + 1) probability distributions on which it is based. In the multifurcating case, we show that Gibbs fragmentation trees are associated with the twoparameter Poisson–Dirichlet models for exchangeable random partitions of N, with an extended parameter range 0 ≤ α ≤ 1, θ ≥−2α and α<0, θ =−mα, m ∈ N.
Prediction Rules for Exchangeable Sequences Related to Species Sampling
 IN PROCESSOR DESIGN. MASTER’S THESIS. LM ERICSSON 2000
, 1998
"... Suppose an exchangable sequence with values in a nice measurable space S admits a prediction rule of the following form: given the first n terms of the sequence, the next term equals the jth distinct value observed so far with probability pj;n , for j = 1; 2; : : :, and otherwise is a new value wit ..."
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Cited by 9 (1 self)
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Suppose an exchangable sequence with values in a nice measurable space S admits a prediction rule of the following form: given the first n terms of the sequence, the next term equals the jth distinct value observed so far with probability pj;n , for j = 1; 2; : : :, and otherwise is a new value with distribution for some probability measure on S with no atoms. Then the pj;n depend only on the partitition of the first n integers induced by the first n values of the sequence. All possible distributions for such an exchangeable sequence are characterized in terms of constraints on the pj;n and in terms of their de Finetti representations.
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.
Limit shapes of Gibbs distributions on the set of integer partitions: The expansive case
, 2008
"... We find limit shapes for a family of multiplicative measures on the set of partitions, induced by exponential generating functions with expansive parameters, ak ∼ Ck p−1, k → ∞, p> 0, where C is a positive constant. The measures considered are associated with the generalized MaxwellBoltzmann models ..."
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Cited by 2 (1 self)
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We find limit shapes for a family of multiplicative measures on the set of partitions, induced by exponential generating functions with expansive parameters, ak ∼ Ck p−1, k → ∞, p> 0, where C is a positive constant. The measures considered are associated with the generalized MaxwellBoltzmann models in statistical mechanics, reversible coagulationfragmentation processes and combinatorial structures, known as assemblies. We prove a central limit theorem for fluctuations of a properly scaled partition chosen randomly according to the above measure, from its limit shape. We demonstrate that when the component size passes beyond the threshold value, the independence of numbers of components transforms into their conditional independence (given their masses). Among other things, the paper also discusses, in a general setting, the interplay between limit shape, threshold and gelation. 1
DISTRIBUTION MEASURES OF THE MEAN VALUE FOR CERTAIN B:ANDOM
"... Let r be a probability measure on [0,1]. We consider a generalization of the classic Dirichlet process the random probability measure F = ~ P~Sx~, where X = {Xi} is a sequence of independent random variables with the common distribution r and P = {P~} is independent of X and has the twoparameter ..."
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Let r be a probability measure on [0,1]. We consider a generalization of the classic Dirichlet process the random probability measure F = ~ P~Sx~, where X = {Xi} is a sequence of independent random variables with the common distribution r and P = {P~} is independent of X and has the twoparameter PoissonDirichlet distribution PD(c~, ~) on the unit simplex. The main result is the formula connecting the distribution I ~ of the random mean value f xdF(x) with the parameter measure r. Bibliography: 12 titles.