Results 1 
7 of
7
Generalized weighted Chinese restaurant processes for species sampling mixture models
 STATISTICA SINICA
, 2003
"... The class of species sampling mixture models is introduced as an extension of semiparametric models based on the Dirichlet process to models based on the general class of species sampling priors, or equivalently the class of all exchangeable urn distributions. Using Fubini calculus in conjunction ..."
Abstract

Cited by 86 (11 self)
 Add to MetaCart
The class of species sampling mixture models is introduced as an extension of semiparametric models based on the Dirichlet process to models based on the general class of species sampling priors, or equivalently the class of all exchangeable urn distributions. Using Fubini calculus in conjunction with Pitman (1995, 1996), we derive characterizations of the posterior distribution in terms of a posterior partition distribution that extend the results of Lo (1984) for the Dirichlet process. These results provide a better understanding of models and have both theoretical and practical applications. To facilitate the use of our models we generalize the work in Brunner, Chan, James and Lo (2001) by extending their weighted Chinese restaurant (WCR) Monte Carlo procedure, an i.i.d. sequential importance sampling (SIS) procedure for approximating posterior mean functionals based on the Dirichlet process, to the case of approximation of mean functionals and additionally their posterior laws in species sampling mixture models. We also discuss collapsed Gibbs sampling, Pólya urn Gibbs sampling and a Pólya urn SIS scheme. Our framework allows for numerous applications, including multiplicative counting process models subject to weighted gamma processes, as well as nonparametric and semiparametric hierarchical models based on the Dirichlet process, its twoparameter extension, the PitmanYor process and finite dimensional Dirichlet priors.
Poisson process partition calculus with an application to Bayesian Levy moving averages
, 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 ..."
Abstract

Cited by 57 (13 self)
 Add to MetaCart
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
QuasiInvariance of the gamma Process and Multiplicative Properties of the PoissonDirichlet Measures
, 1999
"... . In this paper we describe new fundamental properties of the law P \Gamma of the classical gamma process and related properties of the PoissonDirichlet measures PD(`). We prove the quasiinvariance of the measure P \Gamma with respect to an infinitedimensional multiplicative group (the fact ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
(Show Context)
. In this paper we describe new fundamental properties of the law P \Gamma of the classical gamma process and related properties of the PoissonDirichlet measures PD(`). We prove the quasiinvariance of the measure P \Gamma with respect to an infinitedimensional multiplicative group (the fact first discovered in [GGV83]) and the MarkovKrein identity as corollaries of the formula for the Laplace transform of P \Gamma . The quasiinvariance of the measure P \Gamma allows us to obtain new quasiinvariance properties of the measure PD(`). The corresponding invariance properties hold for oefinite analogues of P \Gamma and PD(`). We also show that the measure P \Gamma can be considered as a limit of measures corresponding to the ffstable L'evy processes when parameter ff tends to zero. Our approach is based on simultaneous considering the gamma process (especially its Laplace transform) and its simplicial part  the PoissonDirichlet measures. Quasiinvariance du proces...
Distributions of linear functionals of the two parameter Poisson–Dirichlet random measures
 ANN. APPL. PROBAB
, 2008
"... The present paper provides exact expressions for the probability distributions of linear functionals of the twoparameter Poisson– Dirichlet process PD(α,θ). We obtain distributional results yielding exact forms for density functions of these functionals. Moreover, several interesting integral ident ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
(Show Context)
The present paper provides exact expressions for the probability distributions of linear functionals of the twoparameter Poisson– Dirichlet process PD(α,θ). We obtain distributional results yielding exact forms for density functions of these functionals. Moreover, several interesting integral identities are obtained by exploiting a correspondence between the mean of a Poisson–Dirichlet process and the mean of a suitable Dirichlet process. Finally, some distributional characterizations in terms of mixture representations are proved. The usefulness of the results contained in the paper is demonstrated by means of some illustrative examples. Indeed, our formulae are relevant to occupation time phenomena connected with Brownian motion and more general Bessel processes, as well as to models arising in Bayesian nonparametric statistics.
The twoparameter PoissonDirichlet point process
, 2007
"... The twoparameter PoissonDirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (i.e., the random point process obtai ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
(Show Context)
The twoparameter PoissonDirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (i.e., the random point process obtained by regarding the masses as points in the positive real line) is given in terms of the correlation functions. Relying on this, we apply the theory of point processes to reveal mathematical structure of the twoparameter PoissonDirichlet distribution. Also, developing the Laplace transform approach due to Pitman and Yor, we will be able to extend several results previously known for the oneparameter case, and the MarkovKrein identity for the generalized Dirichlet process is discussed from a point of view of functional analysis based on the twoparameter PoissonDirichlet distribution. 1
Distributional properties of means of random probability measures
, 2009
"... The present paper provides a review of the results concerning distributional properties of means of random probability measures. Our interest in this topic has originated from inferential problems in Bayesian Nonparametrics. Nonetheless, it is worth noting that these random quantities play an import ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
The present paper provides a review of the results concerning distributional properties of means of random probability measures. Our interest in this topic has originated from inferential problems in Bayesian Nonparametrics. Nonetheless, it is worth noting that these random quantities play an important role in seemingly unrelated areas of research. In fact, there is a wealth of contributions both inthe statistics and in theprobability literature that we try to summarize in a unified framework. Particular attention is devoted to means of the Dirichlet process given the relevance of the Dirichlet process in Bayesian Nonparametrics. We then present a number of recent contributions concerning means of more general random probability measures and highlight connections with the moment problem, combinatorics, special functions, excursions of stochastic processes and statistical physics.
A DIFFERENTIAL MODEL FOR THE DEFORMATION OF THE PLANCHEREL GROWTH PROCESS
, 706
"... Abstract. In the present paper we construct and solve a differential model for the qanalog of the Plancherel growth process. The construction is based on a deformation of the MakrovKrein correspondence between continual diagrams and probability distributions. 1. ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
Abstract. In the present paper we construct and solve a differential model for the qanalog of the Plancherel growth process. The construction is based on a deformation of the MakrovKrein correspondence between continual diagrams and probability distributions. 1.