... In this paper we present two general types of Gibbs samplers that can be used to fit posteriors of Bayesian hierarchical models based on stick-breaking 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 stick-breaking 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 non-experts to use.

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.

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 Levy--Ito integral representation is precisely established. Four series representations of a gamma process are given as illustrations of these methods. 1 From L evy--It 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 right--continuous and have left--hand limits (abbreviated as rcll). By the Levy--Ito 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))...

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 tailor-made 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

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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 quasi-invariance 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 sigma-finite measure (quasi-Lebesgue) 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 Poisson-Dirichlet measures. Keywords. Lévy processes, gamma process, stable processes, Poisson–Dirichlet measures, multiplicative quasi-invariance, quasi-Lebesgue measure, Markov–Krein identity.

. In this paper we describe new fundamental properties of the law P \Gamma of the classical gamma process and related properties of the Poisson--Dirichlet measures PD(`). We prove the quasi-invariance of the measure P \Gamma with respect to an infinite-dimensional multiplicative group (the fact first discovered in [GGV83]) and the Markov--Krein identity as corollaries of the formula for the Laplace transform of P \Gamma . The quasi-invariance of the measure P \Gamma allows us to obtain new quasi-invariance properties of the measure PD(`). The corresponding invariance properties hold for oe-finite 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 ff-stable 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 Poisson--Dirichlet measures. Quasi-invariance du proces...

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 well-known and widely used mixture of a Dirichlet process.

We review the current state of nonparametric Bayesian inference. The discussion follows a list of important statistical inference problems, including density estimation, regression, survival analysis, hierarchical models and model validation. For each inference problem we review relevant nonparametric Bayesian models and approaches including Dirichlet process (DP) models and variations, Polya trees, wavelet based models, neural network models, spline regression, CART, dependent DP models, and model validation with DP and Polya tree extensions of parametric models. 1

Functionals of Poisson processes arise in many statistical problems. They appear in problems involving heavy-tailed distributions in the study of limiting processes, while in Bayesian nonparametric statistics they are used as constructive representations for nonparametric priors. We describe a simple recursive method that is useful for characterizing Poisson process functionals that requires only the use of conditional probability. Applications of this technique to convex hulls, extremes, stable measures, infinitely divisible random variables and Bayesian nonparametric priors are discussed.