Results 1  10
of
23
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 ..."
Abstract

Cited by 213 (17 self)
 Add to MetaCart
... 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.
Poisson/gamma random field models for spatial statistics
 BIOMETRIKA
, 1998
"... Doubly stochastic Bayesian hierarchical models are introduced to account for uncertainty and spatial variation in the underlying intensity measure for point process models. Inhomogeneous gamma process random fields and, more generally, Markov random fields with infinitely divisible distributions are ..."
Abstract

Cited by 47 (12 self)
 Add to MetaCart
Doubly stochastic Bayesian hierarchical models are introduced to account for uncertainty and spatial variation in the underlying intensity measure for point process models. Inhomogeneous gamma process random fields and, more generally, Markov random fields with infinitely divisible distributions are used to construct positively autocorrelated intensity measures for spatial Poisson point processes; these in turn are used to model the number and location of individual events. A data augmentation scheme and Markov chain Monte Carlo numerical methods are employed to generate samples from Bayesian posterior and predictive distributions. The methods are developed in both continuous and discrete settings, and are applied to a problem in forest ecology.
Series representations of Lévy processes from the perspective of point processes
"... Several methods of generating series representations of a Levy process are presented under a unified approach and a new rejection method is introduced in this context. The connection of such representations with the LevyIto integral representation is precisely established. Four series representati ..."
Abstract

Cited by 38 (6 self)
 Add to MetaCart
Several methods of generating series representations of a Levy process are presented under a unified approach and a new rejection method is introduced in this context. The connection of such representations with the LevyIto integral representation is precisely established. Four series representations of a gamma process are given as illustrations of these methods. 1 From L evyIt o to series representations. Introduction. Let fX(t) : t 2 [0; 1]g be a Levy process in R d with the characteristic function given by E exp(iuX(t)) = exp t[iua + Z R d 0 (e iux 1 iuxI(jxj 1)) Q(dx)] (1.1) where a 2 R d and Q is a Levy measure on R d 0 (R d 0 := R d n f0g). Assume that the paths of X are rightcontinuous and have lefthand limits (abbreviated as rcll). By the LevyIto integral representation, a.s. for each t 0, X(t) = ta + Z jxj1 x [(N([0; t]; dx) tQ(dx)] + Z jxj>1 xN([0; t]; dx) (1.2) where N is the process of jumps of X : N(A) = P ft: X(t)6=0g 1f(t; X(t))...
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 ..."
Abstract

Cited by 32 (10 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
Nonparametric adaptive estimation for pure jump Lévy processes
, 2008
"... Abstract. This paper is concerned with nonparametric estimation of the Lévy density of a pure jump Lévy process. The sample path is observed at n discrete instants with fixed sampling interval. We construct a collection of estimators obtained by deconvolution methods and deduced from appropriate est ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
Abstract. This paper is concerned with nonparametric estimation of the Lévy density of a pure jump Lévy process. The sample path is observed at n discrete instants with fixed sampling interval. We construct a collection of estimators obtained by deconvolution methods and deduced from appropriate estimators of the characteristic function and its first derivative. We obtain a bound for the L 2risk, under general assumptions on the model. Then we propose a penalty function that allows to build an adaptive estimator. The risk bound for the adaptive estimator is obtained under additional assumptions on the Lévy density. Examples of models fitting in our framework are described and rates of convergence of the estimator are discussed. June 20, 2008
Normalized random measures driven by increasing additive processes
 Annals of Statistics
"... This paper introduces and studies a new class of nonparametric prior distributions. Random probability distribution functions are constructed via normalization of random measures driven by increasing additive processes. In particular, we present results for the distribution of means under both prior ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
This paper introduces and studies a new class of nonparametric prior distributions. Random probability distribution functions are constructed via normalization of random measures driven by increasing additive processes. In particular, we present results for the distribution of means under both prior and posterior conditions and, via the use of strategic latent variables, undertake a full Bayesian analysis. Our class of priors includes the wellknown and widely used mixture of a Dirichlet process.
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 7 (3 self)
 Add to MetaCart
. 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...
Distinguished properties of the gamma processes and related properties
, 2000
"... We study fundamental properties of the gamma process and their relation to various topics such as Poisson–Dirichlet measures and stable processes. We prove the quasiinvariance of the gamma process with respect to a large group of linear transformations. We also show that it is a renormalized limit ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
We study fundamental properties of the gamma process and their relation to various topics such as Poisson–Dirichlet measures and stable processes. We prove the quasiinvariance of the gamma process with respect to a large group of linear transformations. We also show that it is a renormalized limit of the stable processes and has an equivalent sigmafinite measure (quasiLebesgue) with important invariance properties. New properties of the gamma process can be applied to the Poisson—Dirichlet measures. We also emphasize the deep similarity between the gamma process and the Brownian motion. The connection of the above topics makes more transparent some old and new facts about stable and gamma processes, and the PoissonDirichlet measures.
Models beyond the Dirichlet process
 Bayesian Nonparametrics in Practice, CUP
, 2009
"... www.carloalberto.org/working_papers © 2009 by Antonio Lijoi and Igor Prünster. Any opinions expressed here are those of the authors and not those of the Collegio Carlo Alberto. Models beyond the Dirichlet process ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
www.carloalberto.org/working_papers © 2009 by Antonio Lijoi and Igor Prünster. Any opinions expressed here are those of the authors and not those of the Collegio Carlo Alberto. Models beyond the Dirichlet process