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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 ..."
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Cited by 237 (17 self)
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... 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.
Bayesian density regression
 JOURNAL OF THE ROYAL STATISTICAL SOCIETY B
, 2007
"... This article considers Bayesian methods for density regression, allowing a random probability distribution to change flexibly with multiple predictors. The conditional response distribution is expressed as a nonparametric mixture of parametric densities, with the mixture distribution changing acc ..."
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Cited by 44 (24 self)
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This article considers Bayesian methods for density regression, allowing a random probability distribution to change flexibly with multiple predictors. The conditional response distribution is expressed as a nonparametric mixture of parametric densities, with the mixture distribution changing according to location in the predictor space. A new class of priors for dependent random measures is proposed for the collection of random mixing measures at each location. The conditional prior for the random measure at a given location is expressed as a mixture of a Dirichlet process (DP) distributed innovation measure and neighboring random measures. This specification results in a coherent prior for the joint measure, with the marginal random measure at each location being a finite mixture of DP basis measures. Integrating out the infinitedimensional collection of mixing measures, we obtain a simple expression for the conditional distribution of the subjectspecific random variables, which generalizes the Pólya urn scheme. Properties are considered and a simple Gibbs sampling algorithm is developed for posterior computation. The methods are illustrated using simulated data examples and epidemiologic studies.
The Block Diagonal Infinite Hidden Markov Model
"... The Infinite Hidden Markov Model (IHMM) extends hidden Markov models to have a countably infinite number of hidden states (Beal et al., 2002; Teh et al., 2006). We present a generalization of this framework that introduces nearly blockdiagonal structure in the transitions between the hidden states, ..."
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Cited by 3 (2 self)
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The Infinite Hidden Markov Model (IHMM) extends hidden Markov models to have a countably infinite number of hidden states (Beal et al., 2002; Teh et al., 2006). We present a generalization of this framework that introduces nearly blockdiagonal structure in the transitions between the hidden states, where blocks correspond to “subbehaviors” exhibited by data sequences. In identifying such structure, the model classifies, or partitions, sequence data according to these subbehaviors in an unsupervised way. We present an application of this model to artificial data, a video gesture classification task, and a musical theme labeling task, and show that components of the model can also be applied to graph segmentation. 1
Modeling topic hierarchies with the recursive chinese restaurant process
 In Proceedings of the 21st ACM international conference on Information and knowledge management, 783–792. ACM
, 2012
"... Topic models such as latent Dirichlet allocation (LDA) and hierarchical Dirichlet processes (HDP) are simple solutions to discover topics from a set of unannotated documents. While they are simple and popular, a major shortcoming of LDA and HDP is that they do not organize the topics into a hierarc ..."
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Topic models such as latent Dirichlet allocation (LDA) and hierarchical Dirichlet processes (HDP) are simple solutions to discover topics from a set of unannotated documents. While they are simple and popular, a major shortcoming of LDA and HDP is that they do not organize the topics into a hierarchical structure which is naturally found in many datasets. We introduce the recursive Chinese restaurant process (rCRP) and a nonparametric topic model with rCRP as a prior for discovering a hierarchical topic structure with unbounded depth and width. Unlike previous models for discovering topic hierarchies, rCRP allows the documents to be generated from a mixture over the entire set of topics in the hierarchy. We apply rCRP to a corpus of New York Times articles, a dataset of MovieLens ratings, and a set of Wikipedia articles and show the discovered topic hierarchies. We compare the predictive power of rCRP with LDA, HDP, and nested Chinese restaurant process (nCRP) using heldout likelihood to show that rCRP outperforms the others. We suggest two metrics that quantify the characteristics of a topic hierarchy to compare the discovered topic hierarchies of rCRP and nCRP. The results show that rCRP discovers a hierarchy in which the topics become more specialized toward the leaves, and topics in the immediate family exhibit more affinity than topics beyond the immediate family.
MEASURING EXPECTATIONS IN OPTIONS MARKETS: AN APPLICATION TO THE S&P500 INDEX
, 2008
"... Extracting market expectations has always been an important issue when making national policies and investment decisions in financial markets. In option markets, the most popular way has been to extract implied volatilities to assess the future variability of the underlying with the use of the Black ..."
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Extracting market expectations has always been an important issue when making national policies and investment decisions in financial markets. In option markets, the most popular way has been to extract implied volatilities to assess the future variability of the underlying with the use of the Black & Scholes formula. In this manuscript, we propose a novel way to extract the whole time varying distribution of the market implied asset price from option prices. We use a Bayesian nonparametric method that makes use of the Sethuraman representation for Dirichlet processes to take into account the evolution of distributions in time. As an illustration, we present the analysis of options on the S&P500 index.
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.
StickBreaking Beta Processes and the Poisson Process
"... We show that the stickbreaking construction of the beta process due to Paisley et al. (2010) can be obtained from the characterization of the beta process as a Poisson process. Specifically, we show that the mean measure of the underlying Poisson process is equal to that of the beta process. We use ..."
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We show that the stickbreaking construction of the beta process due to Paisley et al. (2010) can be obtained from the characterization of the beta process as a Poisson process. Specifically, we show that the mean measure of the underlying Poisson process is equal to that of the beta process. We use this underlying representation to derive error bounds on truncated beta processes that are tighter than those in the literature. We also develop a new MCMC inference algorithm for beta processes, based in part on our new Poisson process construction. 1
Bayesian Dynamic Modeling of Latent Trait
 Distributions,” Biostatistics
, 2006
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Bayesian Modelling Strategies for Spatially Varying Regression Coefficients: a Multivariate Perspective for
"... This paper considers modelling spatially varying regression effects for multivariate mortality count outcomes. Alternative approaches to spatial regression heterogeneity are considered: the multivariate normal conditional autoregressive (MCAR) model is contrasted with a flexible set of priors based ..."
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This paper considers modelling spatially varying regression effects for multivariate mortality count outcomes. Alternative approaches to spatial regression heterogeneity are considered: the multivariate normal conditional autoregressive (MCAR) model is contrasted with a flexible set of priors based on the multiple membership approach. These include spatial factor priors and a nonparametric approach based on the Dirichlet Process. A case study considers varying regression effects for a bivariate suicide outcome, namely male and female suicides in 354 English local authorities with social deprivation, social fragmentation and rurality as predictors.