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84
Kernel stickbreaking processes
, 2007
"... Summary. This article proposes a class of kernel stickbreaking processes (KSBP) for uncountable collections of dependent random probability measures. The KSBP is constructed by first introducing an infinite sequence of random locations. Independent random probability measures and betadistributed ..."
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Cited by 74 (17 self)
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Summary. This article proposes a class of kernel stickbreaking processes (KSBP) for uncountable collections of dependent random probability measures. The KSBP is constructed by first introducing an infinite sequence of random locations. Independent random probability measures and betadistributed random weights are assigned to each location. Predictordependent random probability measures are then constructed by mixing over the locations, with stickbreaking probabilities expressed as a kernel multiplied by the beta weights. Some theoretical properties of the KSBP are described, including a covariatedependent prediction rule. A retrospective MCMC algorithm is developed for posterior computation, and the methods are illustrated using a simulated example and an epidemiologic application.
THE NESTED DIRICHLET PROCESS
"... In multicenter studies, subjects in different centers may have different outcome distributions. This article is motivated by the problem of nonparametric modeling of these distributions, borrowing information across centers while also allowing centers to be clustered. Starting with a stickbreakin ..."
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Cited by 70 (4 self)
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In multicenter studies, subjects in different centers may have different outcome distributions. This article is motivated by the problem of nonparametric modeling of these distributions, borrowing information across centers while also allowing centers to be clustered. Starting with a stickbreaking representation of the Dirichlet process (DP), we replace the random atoms with random probability measures drawn from a DP. This results in a nested Dirichlet process (nDP) prior, which can be placed on the collection of distributions for the different centers, with centers drawn from the same DP component automatically clustered together. Theoretical properties are discussed, and an efficient MCMC algorithm is developed for computation. The methods are illustrated using a simulation study and an application to quality of care in US hospitals.
TreeStructured Stick Breaking for Hierarchical Data
"... Many data are naturally modeled by an unobserved hierarchical structure. In this paper we propose a flexible nonparametric prior over unknown data hierarchies. The approach uses nested stickbreaking processes to allow for trees of unbounded width and depth, where data can live at any node and are i ..."
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Cited by 50 (8 self)
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Many data are naturally modeled by an unobserved hierarchical structure. In this paper we propose a flexible nonparametric prior over unknown data hierarchies. The approach uses nested stickbreaking processes to allow for trees of unbounded width and depth, where data can live at any node and are infinitely exchangeable. One can view our model as providing infinite mixtures where the components have a dependency structure corresponding to an evolutionary diffusion down a tree. By using a stickbreaking approach, we can apply Markov chain Monte Carlo methods based on slice sampling to perform Bayesian inference and simulate from the posterior distribution on trees. We apply our method to hierarchical clustering of images and topic modeling of text data. 1
Compressive Sensing on Manifolds Using a Nonparametric Mixture of Factor Analyzers: Algorithm and Performance Bounds 1
"... Nonparametric Bayesian methods are employed to constitute a mixture of lowrank Gaussians, for data x ∈ RN that are of high dimension N but are constrained to reside in a lowdimensional subregion of RN. The number of mixture components and their rank are inferred automatically from the data. The re ..."
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Cited by 46 (17 self)
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Nonparametric Bayesian methods are employed to constitute a mixture of lowrank Gaussians, for data x ∈ RN that are of high dimension N but are constrained to reside in a lowdimensional subregion of RN. The number of mixture components and their rank are inferred automatically from the data. The resulting algorithm can be used for learning manifolds and for reconstructing signals from manifolds, based on compressive sensing (CS) projection measurements. The statistical CS inversion is performed analytically. We derive the required number of CS random measurements needed for successful reconstruction, based on easily computed quantities, drawing on block–sparsity properties. The proposed methodology is validated on several synthetic and real datasets. I.
Slice Sampling Mixture Models
"... We propose a more efficient version of the slice sampler for Dirichlet process mixture models described by Walker (2007). This sampler allows the fitting of infinite mixture models with a wide–range of prior specification. To illustrate this flexiblity we develop a new nonparametric prior for mixtur ..."
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Cited by 42 (2 self)
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We propose a more efficient version of the slice sampler for Dirichlet process mixture models described by Walker (2007). This sampler allows the fitting of infinite mixture models with a wide–range of prior specification. To illustrate this flexiblity we develop a new nonparametric prior for mixture models by normalizing an infinite sequence of independent positive random variables and show how the slice sampler can be applied to make inference in this model. Two submodels are studied in detail. The first one assumes that the positive random variables are Gamma distributed and the second assumes that they are inverse– Gaussian distributed. Both priors have two hyperparameters and we consider their effect on the prior distribution of the number of occupied clusters in a sample. Extensive computational comparisons with alternative ”conditional” simulation techniques for mixture models using the standard Dirichlet process prior and our new prior are made. The properties of the new prior are illustrated on a density estimation problem.
A general framework for the parametrization of hierarchical models.
 Statistical Science,
, 2007
"... ..."
Nonparametric Bayesian models through probit stickbreaking processes
"... Summary. We describe a novel class of Bayesian nonparametric priors based on stickbreaking constructions where the weights of the process are constructed as probit transformations of normal random variables. We show that these priors are extremely flexible, allowing us to generate a great variety of ..."
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Cited by 22 (6 self)
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Summary. We describe a novel class of Bayesian nonparametric priors based on stickbreaking constructions where the weights of the process are constructed as probit transformations of normal random variables. We show that these priors are extremely flexible, allowing us to generate a great variety of models while preserving computational simplicity. Particular emphasis is placed on the construction of rich temporal and spatial processes, which are applied to two problems in finance and ecology.
MCMC for normalized random measure mixture models
, 2013
"... This paper concerns the use of Markov chain Monte Carlo methods for posterior sampling in Bayesian nonparametric mixture models with normalized random measure priors. Making use of some recent posterior characterizations for the class of normalized random measures, we propose novel Markov chain Mon ..."
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Cited by 20 (9 self)
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This paper concerns the use of Markov chain Monte Carlo methods for posterior sampling in Bayesian nonparametric mixture models with normalized random measure priors. Making use of some recent posterior characterizations for the class of normalized random measures, we propose novel Markov chain Monte Carlo methods of both marginal type and conditional type. The proposed marginal samplers are generalizations of Neal’s wellregarded Algorithm 8 for Dirichlet process mixture models, whereas the conditional sampler is a variation of those recently introduced in the literature. For both the marginal and conditional methods, we consider as a running example a mixture model with an underlying normalized generalized Gamma process prior, and describe comparative simulation results demonstrating the efficacies of the proposed methods.
Negative Binomial Process Count and Mixture Modeling
, 2013
"... The seemingly disjoint problems of count and mixture modeling are united under the negative binomial (NB) process. A gamma process is employed to model the rate measure of a Poisson process, whose normalization provides a random probability measure for mixture modeling and whose marginalization lead ..."
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Cited by 17 (10 self)
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The seemingly disjoint problems of count and mixture modeling are united under the negative binomial (NB) process. A gamma process is employed to model the rate measure of a Poisson process, whose normalization provides a random probability measure for mixture modeling and whose marginalization leads to an NB process for count modeling. A draw from the NB process consists of a Poisson distributed finite number of distinct atoms, each of which is associated with a logarithmic distributed number of data samples. We reveal relationships between various count and mixturemodeling distributions and construct a Poissonlogarithmic bivariate distribution that connects the NB and Chinese restaurant table distributions. Fundamental properties of the models are developed, and we derive efficient Bayesian inference. It is shown that with augmentation and normalization, the NB process and gammaNB process can be reduced to the Dirichlet process and hierarchical Dirichlet process, respectively. These relationships highlight theoretical, structural and computational advantages of the NB process. A variety of NB processes, including the betageometric, betaNB, markedbetaNB, markedgammaNB and zeroinflatedNB processes, with distinct sharing mechanisms, are also constructed. These models are applied to topic modeling, with connections made to existing algorithms under Poisson factor analysis. Example results show the importance of inferring both the NB dispersion and probability parameters.
The Gaussian Process Density Sampler
"... We present the Gaussian Process Density Sampler (GPDS), an exchangeable generative model for use in nonparametric Bayesian density estimation. Samples drawn from the GPDS are consistent with exact, independent samples from a fixed density function that is a transformation of a function drawn from a ..."
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Cited by 15 (4 self)
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We present the Gaussian Process Density Sampler (GPDS), an exchangeable generative model for use in nonparametric Bayesian density estimation. Samples drawn from the GPDS are consistent with exact, independent samples from a fixed density function that is a transformation of a function drawn from a Gaussian process prior. Our formulation allows us to infer an unknown density from data using Markov chain Monte Carlo, which gives samples from the posterior distribution over density functions and from the predictive distribution on data space. We can also infer the hyperparameters of the Gaussian process. We compare this density modeling technique to several existing techniques on a toy problem and a skullreconstruction task. 1