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17
Sequential Importance Sampling for Nonparametric Bayes Models: The Next Generation
 Journal of Statistics
, 1998
"... this paper, we exploit the similarities between the Gibbs sampler and the SIS, bringing over the improvements for Gibbs sampling algorithms to the SIS setting for nonparametric Bayes problems. These improvements result in an improved sampler and help satisfy questions of Diaconis (1995) pertaining t ..."
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Cited by 68 (5 self)
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this paper, we exploit the similarities between the Gibbs sampler and the SIS, bringing over the improvements for Gibbs sampling algorithms to the SIS setting for nonparametric Bayes problems. These improvements result in an improved sampler and help satisfy questions of Diaconis (1995) pertaining to convergence. Such an effort can see wide applications in many other problems related to dynamic systems where the SIS is useful (Berzuini et al. 1996; Liu and Chen 1996). Section 2 describes the specific model that we consider. For illustration we focus discussion on the betabinomial model, although the methods are applicable to other conjugate families. In Section 3, we describe the first generation of the SIS and Gibbs sampler in this context, and present the necessary conditional distributions upon which the techniques rely. Section 4 describes the alterations that create the second generation techniques, and provides specific algorithms for the model we consider. Section 5 presents a comparison of the techniques on a large set of data. Section 6 provides theory that ensures the proposed methods work and that is generally applicable to many other problems using importance sampling approaches. The final section presents discussion. 2 The Model
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 ..."
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Cited by 56 (1 self)
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: 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...
Bayesian Multiple Comparisons Using Dirichlet Process Priors
 Journal of the American Statistical Association
, 1996
"... We consider the problem of multiple comparisons from a Bayesian viewpoint. The family of Dirichlet process priors is applied in the form of baseline prior/likelihood combinations, to obtain posterior probabilities for various hypotheses. The baseline prior/likelihood combinations considered here are ..."
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Cited by 15 (0 self)
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We consider the problem of multiple comparisons from a Bayesian viewpoint. The family of Dirichlet process priors is applied in the form of baseline prior/likelihood combinations, to obtain posterior probabilities for various hypotheses. The baseline prior/likelihood combinations considered here are beta/binomial, normal/inverted gamma with equal variances and a hierarchical nonconjugate normal/inverted gamma prior on treatment means. The prior probabilities of the hypotheses depend directly on the concentration parameter of the Dirichlet process prior. The problem is analytically intractable; we use Gibbs sampling. The posterior probabilities of the hypotheses are easily obtained as a byproduct in evaluating the marginal posterior distributions of the parameters. The proposed procedure is compared with Duncan's multiple range test and shown to be more powerful under certain alternative hypotheses. Keywords: Gibbs sampling, beta/binomial prior, normal/inverted gamma prior, hierarchica...
Nonparametric Bayes KernelBased Priors for Functional Data Analysis
"... Abstract: We focus on developing nonparametric Bayes methods for collections of dependent random functions, allowing individual curves to vary flexibly while adaptively borrowing information. A prior is proposed, which is expressed as a hierarchical mixture of weighted kernels placed at unknown loca ..."
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Cited by 9 (1 self)
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Abstract: We focus on developing nonparametric Bayes methods for collections of dependent random functions, allowing individual curves to vary flexibly while adaptively borrowing information. A prior is proposed, which is expressed as a hierarchical mixture of weighted kernels placed at unknown locations. The induced prior for any individual function is shown to fall within a reproducing kernel Hilbert space. We allow flexible borrowing of information through the use of a hierarchical Dirichlet process prior for the random locations, along with a functional Dirichlet process for the weights. Theoretical properties are considered and an efficient MCMC algorithm is developed, relying on stickbreaking truncations. The methods are illustrated using simulation examples and an application to reproductive hormone data.
NONPARAMETRIC PRIORS FOR ORDINAL BAYESIAN SOCIAL SCIENCE MODELS: SPECIFICATION AND ESTIMATION
"... A generalized linear mixed model, and ordered probit, is used to estimate levels of stress in presidential political appointees as a means of understanding their surprising short tenures. A Bayesian approach is used, where the random effects are modeled with a Dirichlet process mixture prior, allowi ..."
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Cited by 7 (1 self)
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A generalized linear mixed model, and ordered probit, is used to estimate levels of stress in presidential political appointees as a means of understanding their surprising short tenures. A Bayesian approach is used, where the random effects are modeled with a Dirichlet process mixture prior, allowing for some incorporation of prior information, but retaining some vagueness in the form of the prior. Applications of Bayesian models in the social sciences are typically done with “noninformative” priors, although some use of informed versions exists. There has been disagreement over this, and our approach may be a step in the direction of satisfying both camps. We give a detailed description of the data, show how to implement the model, and describe some interesting conclusions. The model utilizing a nonparametric prior fits better and reveals more information in the data.
A note on the Dirichlet process prior in Bayesian nonparametric inference with partial exchangeability
 Statist. Prob. Letters
, 1997
"... We consider Bayesian nonparametric inference for continuousvalued partially exchangeable data, when the partition of the observations into groups is unknown. This includes changepoint problems and mixture models. As the prior, we consider a mixture of products of Dirichlet processes. We show that ..."
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Cited by 6 (1 self)
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We consider Bayesian nonparametric inference for continuousvalued partially exchangeable data, when the partition of the observations into groups is unknown. This includes changepoint problems and mixture models. As the prior, we consider a mixture of products of Dirichlet processes. We show that the discreteness of the Dirichlet process can have a large effect on inference (posterior distributions and Bayes factors), leading to conclusions that can be different from those that result from a reasonable parametric model. When the observed data are all distinct, the effect of the prior on the posterior is to favor more evenly balanced partitions, and its effect on Bayes factors is to favor more groups. In a hierarchical model with a Dirichlet process as the secondstage prior, the prior can also have a large effect on inference, but in the opposite direction, towards more unbalanced partitions. (~) 1997 Elsevier Science B.V.
Hierarchical Models for Estimating Herd Prevalence and Test Accuracy in the Absence of a GoldStandard
 Journal of Agricultural, Biological and Environmental Statistics
, 2003
"... Introduction While there are currently no worldwide data on the cost of screening programs for animal diseases, about $60 million is currently spent annually in the United States by the United States Department of Agriculture for animal 1 disease surveillance, a major component of which involves d ..."
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Cited by 4 (4 self)
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Introduction While there are currently no worldwide data on the cost of screening programs for animal diseases, about $60 million is currently spent annually in the United States by the United States Department of Agriculture for animal 1 disease surveillance, a major component of which involves disease testing of animals and prevalence estimation across farms (cf. Hanson et al 2000). Useful measures of binary test accuracy are sensitivity and specicity. When a perfect reference test, or goldstandard, is available these parameters are readily estimated. However, a goldstandard is often unavailable, prohibitively expensive or morbidly invasive to perform, and alternate methods of estimating sensitivity and specicity are needed. Hui and Walter (1980) introduced a model that provides consistent maximum likelihood estimates of test sensititivy and specicity for two or more tests and two or more populations. Model identiability is achieved by assuming conditional inde
Defining predictive probability functions for species sampling models
, 2009
"... We review the class of species sampling models (SSM). In particular, we investigate the relation between the exchangeable partition probability function (EPPF) and the predictive probability function (PPF). It is straightforward to define a PPF from an EPPF, but the converse is not necessarily true. ..."
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Cited by 3 (0 self)
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We review the class of species sampling models (SSM). In particular, we investigate the relation between the exchangeable partition probability function (EPPF) and the predictive probability function (PPF). It is straightforward to define a PPF from an EPPF, but the converse is not necessarily true. In this paper, we introduce the notion of putative PPFs and show novel conditions for a putative PPF to define an EPPF. We show that all possible PPFs in a certain class have to define (unnormalized) probabilities for cluster membership that are linear in cluster size. We give a new necessary and sufficient condition for arbitrary putative PPFs to define an EPPF. Finally we show posterior inference for a large class of SSMs with a PPF that is not linear in cluster size and discuss a numerical method to derive its PPF. Key words and phrases: Species sampling Prior, Exchangeable partition probability functions, Prediction probability functions. 1 1
Posterior Integration in Dynamic Models
 Computing Science and Statistics
, 1992
"... The analysis of general dynamic models involves a sequence of posterior distributions corresponding to the subsequent stages of the dynamic model. In the absence of normal/linear structure numerical integration schemes are required to estimate features of these posterior distributions. This paper r ..."
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Cited by 2 (0 self)
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The analysis of general dynamic models involves a sequence of posterior distributions corresponding to the subsequent stages of the dynamic model. In the absence of normal/linear structure numerical integration schemes are required to estimate features of these posterior distributions. This paper reviews some previously suggested Monte Carlo based algorithms and suggests a new scheme which makes use of a Metropolis type algorithm to propagate a Monte Carlo sample simulated from the initial prior distribution through all stages of the dynamic model. For each of the posterior distributions in the dynamic model, the algorithm makes a Monte Carlo sample available which allows then to estimate posterior integrals as desired. Before proceeding to the analysis at time t, the algorithm requires reconstruction of the posterior distribution corresponding to period t \Gamma 1. This is solved by an implementation of a mixture of Dirichlet process model, making use of the already available Monte C...
Nonparametric empirical Bayes for the Dirichlet process mixture model
 Statistics and Computing
, 2004
"... The Dirichlet process prior allows flexible nonparametric mixture modeling. The number of mixture components is not specified in advance and can grow as new data come in. However, the behavior of the model is sensitive to the choice of the parameters, including an infinitedimensional distribution ..."
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Cited by 2 (0 self)
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The Dirichlet process prior allows flexible nonparametric mixture modeling. The number of mixture components is not specified in advance and can grow as new data come in. However, the behavior of the model is sensitive to the choice of the parameters, including an infinitedimensional distributional parameter G0 . Most previous applications have either fixed G0 as a member of a parametric family or treated G0 in a Bayesian fashion, using parametric prior specifications. In contrast, we have developed an adaptive nonparametric method for constructing smooth estimates of G0 . We combine this method with a technique for estimating #, the other Dirichlet process parameter, that is inspired by an existing characterization of its maximumlikelihood estimator. Together, these estimation procedures yield a flexible empirical Bayes treatment of Dirichlet process mixtures. Such a treatment is useful in situations where smooth point estimates of G0 are of intrinsic interest, or where the structure of G0 cannot be conveniently modeled with the usual parametric prior families. Analysis of simulated and realworld datasets illustrates the robustness of this approach.