Results 1  10
of
29
Generalized weighted Chinese restaurant processes for species sampling mixture models
 Statistica Sinica
, 2003
"... Abstract: 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 conj ..."
Abstract

Cited by 52 (8 self)
 Add to MetaCart
Abstract: 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. Key words and phrases: Dirichlet process, exchangeable partition, finite dimensional Dirichlet prior, twoparameter PoissonDirichlet process, prediction rule, random probability measure, species sampling sequence.
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 ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
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.
Bayesian model based clustering procedures
 Journal of Computational and Graphical Statistics Lo
, 2006
"... This paper establishes a general framework for Bayesian modelbased clustering, in which subset labels are exchangeable, and items are also exchangeable, possibly up to covariate effects. It is rich enough to encompass a variety of existing procedures, including some recently discussed methodologies ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
This paper establishes a general framework for Bayesian modelbased clustering, in which subset labels are exchangeable, and items are also exchangeable, possibly up to covariate effects. It is rich enough to encompass a variety of existing procedures, including some recently discussed methodologies involving stochastic search or hierarchical clustering, but more importantly allows the formulation of clustering procedures that are optimal with respect to a specified loss function. Our focus is on loss functions based on pairwise coincidences, that is, whether pairs of items are clustered into the same subset or not. Optimisation of the posterior expected loss function can be formulated as a binary integer programming problem, which can be readily solved, for example by the simplex method, when clustering a modest number of items, but quickly becomes impractical as problem scale increases. To combat this, a new heuristic itemswapping algorithm is introduced. This performs well in our numerical experiments, on both simulated and real data examples. The paper includes a comparison of the statistical performance of the (approximate) optimal clustering with earlier methods that are modelbased but ad hoc in their detailed definition.
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 ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
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.
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 8 (5 self)
 Add to MetaCart
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.
Distributions of functionals of the two parameter PoissonDirichlet process
, 2006
"... The present paper provides exact expressions for the probability distribution of linear functionals of the two–parameter Poisson–Dirichlet process PD(α, θ). Distributional results that follow from the application of an inversion formula for a (generalized) Cauchy– Stieltjes transform are achieved. ..."
Abstract

Cited by 7 (6 self)
 Add to MetaCart
The present paper provides exact expressions for the probability distribution of linear functionals of the two–parameter Poisson–Dirichlet process PD(α, θ). Distributional results that follow from the application of an inversion formula for a (generalized) Cauchy– Stieltjes transform are achieved. Moreover, several interesting integral identities are obtained by exploiting a correspondence between the mean functional of a Poisson–Dirichlet process and the mean functional of a suitable Dirichlet process. Finally, some distributional characterizations in terms of mixture representations are illustrated. 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.
Bayesian nonparametric estimator derived from conditional Gibbs structures. Annals of Applied Probability
 J. Phys. A: Math. Gen
, 2008
"... We consider discrete nonparametric priors which induce Gibbstype exchangeable random partitions and investigate their posterior behavior in detail. In particular, we deduce conditional distributions and the corresponding Bayesian nonparametric estimators, which can be readily exploited for predictin ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
We consider discrete nonparametric priors which induce Gibbstype exchangeable random partitions and investigate their posterior behavior in detail. In particular, we deduce conditional distributions and the corresponding Bayesian nonparametric estimators, which can be readily exploited for predicting various features of additional samples. The results provide useful tools for genomic applications where prediction of future outcomes is required. 1. Introduction. Random
A Bayes method for a monotone hazard rate via Spaths
 Ann. Statist
, 2006
"... A class of random hazard rates, that is defined as a mixture of an indicator kernel convoluted with a completely random measure, is of interest. We provide an explicit characterization of the posterior distribution of this mixture hazard rate model via a finite mixture of Spaths. A closed and tract ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
A class of random hazard rates, that is defined as a mixture of an indicator kernel convoluted with a completely random measure, is of interest. We provide an explicit characterization of the posterior distribution of this mixture hazard rate model via a finite mixture of Spaths. A closed and tractable Bayes estimator for the hazard rate is derived to be a finite sum over Spaths. The path characterization or the estimator is proved to be a RaoBlackwellization of an existing partition characterization or partitionsum estimator. This accentuates the importance of Spath in Bayesian modeling of monotone hazard rates. An efficient Markov chain Monte Carlo (MCMC) method is proposed to approximate this class of estimates. It is shown that Spath characterization also exists in modeling with covariates by a proportional hazard model, and the proposed algorithm again applies. Numerical results of the method are given to demonstrate its practicality and effectiveness.
Analysis of a class of likelihood based continuous time stochastic volatility models including OrnsteinUhlenbeck models in financial economics. arXiv:math.ST/0503055
, 2005
"... In a series of recent papers BarndorffNielsen and Shephard introduce an attractive class of continuous time stochastic volatility models for financial assets where the volatility processes are functions of positive OrnsteinUhlenbeck(OU) processes. These models are known to be substantially more fl ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
In a series of recent papers BarndorffNielsen and Shephard introduce an attractive class of continuous time stochastic volatility models for financial assets where the volatility processes are functions of positive OrnsteinUhlenbeck(OU) processes. These models are known to be substantially more flexible than Gaussian based models. One current problem of this approach is the unavailability of a tractable exact analysis of likelihood based stochastic volatility models for the returns of log prices of stocks. With this point in mind, the likelihood models of BarndorffNielsen and Shephard are viewed as members of a much larger class of models. That is likelihoods based on n conditionally independent Normal random variables whose mean and variance are representable as linear functionals of a common unobserved Poisson random measure. The analysis of these models is facilitated by applying the methods in James (2005, 2002), in particular an Esscher type transform of Poisson random measures; in conjunction with a special case of the WeberSonine formula. It is shown that the marginal likelihood may be expressed in terms of a multidimensional Fouriercosine transform. This yields tractable forms of the likelihood and also allows a full Bayesian posterior analysis of the integrated volatility process. A general formula for the posterior density of the log price given the observed data is derived, which could potentially have applications to option pricing. We also identify tractable subclasses, where inference can be based on a finite number of independent random variables. We close by obtaining explicit expressions for likelihoods incorporating leverage. It is shown that inference does not necessarily require simulation of random measures. Rather, classical numerical integration can be used in the most general cases. 1