Results 1  10
of
15
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 237 (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.
Iterated random functions
 SIAM Review
, 1999
"... Abstract. Iterated random functions are used to draw pictures or simulate large Ising models, among other applications. They offer a method for studying the steady state distribution of a Markov chain, and give useful bounds on rates of convergence in a variety of examples. The present paper surveys ..."
Abstract

Cited by 143 (1 self)
 Add to MetaCart
Abstract. Iterated random functions are used to draw pictures or simulate large Ising models, among other applications. They offer a method for studying the steady state distribution of a Markov chain, and give useful bounds on rates of convergence in a variety of examples. The present paper surveys the field and presents some new examples. There is a simple unifying idea: the iterates of random Lipschitz functions converge if the functions are contracting on the average. 1. Introduction. The
The Consistency of Posterior Distributions in Nonparametric Problems
 Ann. Statist
, 1996
"... We give conditions that guarantee that the posterior probability of every Hellinger... ..."
Abstract

Cited by 89 (4 self)
 Add to MetaCart
(Show Context)
We give conditions that guarantee that the posterior probability of every Hellinger...
More Aspects of Polya Tree Distributions for Statistical Modelling
 Ann. Statist
, 1994
"... : The definition and elementary properties of Polya tree distributions are reviewed. Two theorems are presented showing that Polya trees can be constructed to concentrate arbitrarily closely about any desired pdf, and that Polya tree priors can put positive mass in every relative entropy neighborhoo ..."
Abstract

Cited by 65 (1 self)
 Add to MetaCart
: The definition and elementary properties of Polya tree distributions are reviewed. Two theorems are presented showing that Polya trees can be constructed to concentrate arbitrarily closely about any desired pdf, and that Polya tree priors can put positive mass in every relative entropy neighborhood of every positive density with finite entropy, thereby satisfying a consistency condition. Such theorems are false for Dirichlet processes. Models are constructed combining partially specified Polya trees with other information like monotonicity or unimodality. It is shown how to compute bounds on posterior expectations over the class of all priors with the given specifications. A numerical example is given. A theorem of Diaconis and Freedman about Dirichlet processes is generalized to Polya trees, allowing Polya trees to be the models for errors in regression problems. Finally, empirical Bayes models using Dirichlet processes are generalized to Polya trees. An example from Berry and Chris...
Modeling Regression Error with a Mixture of Polya Trees
 Journal of the American Statistical Association
, 2001
"... We model the error distribution in the standard linear model as a mixture of absolutely continuous Polya trees constrained to have median zero. By considering a mixture, we smooth out the partitioning e ects of a simple Polya tree and the predictive error density has a derivative everywhere except z ..."
Abstract

Cited by 23 (2 self)
 Add to MetaCart
We model the error distribution in the standard linear model as a mixture of absolutely continuous Polya trees constrained to have median zero. By considering a mixture, we smooth out the partitioning e ects of a simple Polya tree and the predictive error density has a derivative everywhere except zero. The error distribution is centered around a standard parametric family of distributions and may therefore be viewed as a generalization of standard models in which important, datadriven features, such as skewness and multimodality, are allowed. By marginalizing the Polya tree exact inference is possible up to MCMC error.
Some Further Developments for StickBreaking Priors: Finite and Infinite Clustering and Classification
 Sankhya Series A
, 2003
"... this paper will be to develop new surrounding theory for the hierarchical model (7) and show how these may be used to develop computational algorithms for computing posterior quantities. Our theoretical contributions include developing key properties for the class of extended stickbreaking measures ..."
Abstract

Cited by 18 (0 self)
 Add to MetaCart
(Show Context)
this paper will be to develop new surrounding theory for the hierarchical model (7) and show how these may be used to develop computational algorithms for computing posterior quantities. Our theoretical contributions include developing key properties for the class of extended stickbreaking measures, which includes establishing a conjugacy property of their random weights to i.i.d sampling, and a characterization of the posterior for the extended stickbreaking prior under i.i.d sampling. See Section 3. These properties then lead us in Section 4 to a general characterization for the posterior of (7). In Section 5 we outline a collapsed Gibbs sampling algorithm and an i.i.d SIS (sequential importance sampling) algorithm that can be used for inference in (7). One important implication is our ability to t the posterior of (6) subject to in nite dimensional stickbreaking measures. The paper begins with a brief discussion of stickbreaking priors in Section 2
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 10 (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.
A recursive method for functionals of Poisson processes
 Bernoulli
, 2002
"... Functionals of Poisson processes arise in many statistical problems. They appear in problems involving heavytailed distributions in the study of limiting processes, while in Bayesian nonparametric statistics they are used as constructive representations for nonparametric priors. We describe a simpl ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Functionals of Poisson processes arise in many statistical problems. They appear in problems involving heavytailed 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.
Parametric and nonparametric Bayesian model specification: A case study involving models for count data
 Computational Statistics & Data Analysis
, 2008
"... Abstract In this paper we present the results of a simulation study to explore the ability of Bayesian parametric and nonparametric models to provide an adequate fit to count data, of the type that would routinely be analyzed parametrically either through fixedeffects or randomeffects Poisson mode ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Abstract In this paper we present the results of a simulation study to explore the ability of Bayesian parametric and nonparametric models to provide an adequate fit to count data, of the type that would routinely be analyzed parametrically either through fixedeffects or randomeffects Poisson models. The context of the study is a randomized controlled trial with two groups (treatment and control). Our nonparametric approach utilizes several modeling formulations based on Dirichlet process priors. We find that the nonparametric models are able to flexibly adapt to the data, to offer rich posterior inference, and to provide, in a variety of settings, more accurate predictive inference than parametric models.