... 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.

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

A common objective in learning a model from data is to recover its network structure, while the model parameters are of minor interest. For example, we may wish to recover regulatory networks from high-throughput data sources. In this paper we examine how Bayesian regularization using a Dirichlet prior over the model parameters affects the learned model structure in a domain with discrete variables. Surprisingly, a weak prior in the sense of smaller equivalent sample size leads to a strong regularization of the model structure (sparse graph) given a sufficiently large data set. In particular, the empty graph is obtained in the limit of a vanishing strength of prior belief. This is diametrically opposite to what one may expect in this limit, namely the complete graph from an (unregularized) maximum likelihood estimate. Since the prior affects the parameters as expected, the prior strength balances a "trade-off" between regularizing the parameters or the structure of the model. We demonstrate the benefits of optimizing this trade-off in the sense of predictive accuracy.

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Widely used parametric generalizedlinear models are, unfortunately,a somewhat limited class of speci � cations. Nonparametric aspects are often introduced to enrich this class, resultingin semiparametricmodels. Focusing on single or k-sample problems,many classical nonparametricapproachesare limited to hypothesistesting. Those that allow estimation are limited to certain functionals of the underlying distributions. Moreover, the associated inference often relies upon asymptotics when nonparametric speci � cations are often most appealing for smaller sample sizes. Bayesian nonparametricapproachesavoid asymptotics but have, to date, been limited in the range of inference. Working with Dirichlet process priors, we overcome the limitations of existing simulation-basedmodel � tting approaches which yield inference that is con � ned to posterior moments of linear functionals of the population distribution.This article provides a computationalapproach to obtain the entire posterior distribution for more general functionals. We illustrate with three applications: investigation of extreme value distributions associated with a single population, comparison of medians in a k-sample problem, and comparison of survival times from different populations under fairly heavy censoring.

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this paper is based on a nonparametric model for the unknown joint distribution for the missing data, the observed covariates and the proxy. This nonparametric distribution defines the measurement error component of the model which relates the missing covariates X with a proxy W . The oxymoron "non-parametric Bayes" refers to a class of flexible mixture distributions. For the likelihood of disease, given covariates, we choose a logistic regression model. By using a parametric disease model and nonparametric exposure model we obtain robust, interpretable, results quantifying the effect of exposure. Some Key Words: Mixture of Dirichlet Processes, Logistic Regression, Measurement Error, Markov chain Monte Carlo, Gibbs Sampling, Metropolis Sampling. 1 Introduction We develop a model and a numerical estimation scheme for a Bayesian approach to inference in case-control studies with errors in covariables. In epidemiology covariates are frequently measured with error substantial enough that it can seriously affect the assessment of the relation between risk factors and disease outcome (Carroll, Spiegelman, Lan, Bailey & Abbott 1984). Although there are situations where it is possible to obtain accurate measurements of the covariate, at a substantially higher cost, for the majority of the observations these measurements will not be available. Nevertheless, the validation group can be used to form a model that indirectly links the error-prone measurement to the disease outcome, through its relation to the error-free measurement. This general topic of measurement error models has received a considerable amount of attention in the frequentist literature (e.g., Fuller, 1987; Stefanski & Carroll, 1990), but until recently has received relatively little attention in the Bayesian li...

Abstract: We propose a method for the analysis of a spatial point pattern, which is assumed to arise as a set of observations from a spatial non-homogeneous Poisson process. The spatial point pattern is observed in a bounded region, which, for most applications, is taken to be a rectangle in the space where the process is defined. The method is based on modeling a density function, defined on this bounded region, that is directly related with the intensity function of the Poisson process. We develop a flexible nonparametric mix-ture model for this density using a bivariate Beta distribution for the mixture kernel and a Dirichlet process prior for the mixing distribution. Using posterior simulation methods, we obtain full inference for the intensity function and any other functional of the process that might be of interest. We discuss applications to problems where inference for clus-tering in the spatial point pattern is of interest. Moreover, we consider applications of the methodology to extreme value analysis problems. We illustrate the modeling approach with three previously published data sets. Two of the data sets are from forestry and consist of locations of trees. The third data set consists of extremes from the Dow Jones index over a period of 1303 days.

The nonparametric Bayesian approach for inference regarding the unknown distribution of a random sample customarily assumes that this distribution is random and arises through Dirichlet process mixing. Previous work within this setting has focused on the mean of the posterior distribution of this random distribution which is the predictive distribution of a future observation given the sample. Our interest here is in learning about other features of this posterior distribution as well as about posteriors associated with functionals of the distribution of the data. We indicate how to do this in the case of linear functionals. An illustration, with a sample from a Gamma distribution, utilizes Dirichlet process mixtures of normals to recover this distribution and its features. Key Words: Dirichlet process, linear functional, posterior distribution, predictive distribution, sampling-based inference. AMS Subject Classifications: 62C10, 62G07. 1 Alan E. Gelfand is Professor and Saurabh Mukh...

We discuss two methods of making nonparametric Bayesian inference on probability measures subject to a partial stochastic ordering. The first method involves a nonparametric prior for a measure on partially ordered latent observations, and the second involves rejection sampling. Computational approaches are discussed for each method, and interpretations of prior and posterior information are discussed. An application is presented in which inference is made on the number of independently segregating quantitative trait loci present in an animal population.