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74
NonParametric Bayesian Dictionary Learning for Sparse Image Representations
"... Nonparametric Bayesian techniques are considered for learning dictionaries for sparse image representations, with applications in denoising, inpainting and compressive sensing (CS). The beta process is employed as a prior for learning the dictionary, and this nonparametric method naturally infers ..."
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Cited by 92 (34 self)
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Nonparametric Bayesian techniques are considered for learning dictionaries for sparse image representations, with applications in denoising, inpainting and compressive sensing (CS). The beta process is employed as a prior for learning the dictionary, and this nonparametric method naturally infers an appropriate dictionary size. The Dirichlet process and a probit stickbreaking process are also considered to exploit structure within an image. The proposed method can learn a sparse dictionary in situ; training images may be exploited if available, but they are not required. Further, the noise variance need not be known, and can be nonstationary. Another virtue of the proposed method is that sequential inference can be readily employed, thereby allowing scaling to large images. Several example results are presented, using both Gibbs and variational Bayesian inference, with comparisons to other stateoftheart approaches.
Retrospective Markov chain Monte Carlo methods for Dirichlet process hierarchical models
 PROC. IEEE
, 2008
"... Inference for Dirichlet process hierarchical models is typically performed using Markov chain Monte Carlo methods, which can be roughly categorised into marginal and conditional methods. The former integrate out analytically the infinitedimensional component of the hierarchical model and sample fro ..."
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Cited by 84 (5 self)
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Inference for Dirichlet process hierarchical models is typically performed using Markov chain Monte Carlo methods, which can be roughly categorised into marginal and conditional methods. The former integrate out analytically the infinitedimensional component of the hierarchical model and sample from the marginal distribution of the remaining variables using the Gibbs sampler. Conditional methods impute the Dirichlet process and update it as a component of the Gibbs sampler. Since this requires imputation of an infinitedimensional process, implementation of the conditional method has relied on finite approximations. In this paper we show how to avoid such approximations by designing two novel Markov chain Monte Carlo algorithms which sample from the exact posterior distribution of quantities of interest. The approximations are avoided by the new technique of retrospective sampling. We also show how the algorithms can obtain samples from functionals of the Dirichlet process. The marginal and the conditional methods are compared and a careful simulation study is included, which involves a nonconjugate model, different datasets and prior specifications.
Nonparametric bayes conditional distribution modeling with variable selection
 Journal of the American Statistical Association
, 2009
"... This article considers methodology for flexibly characterizing the relationship between a response and multiple predictors. Goals are (1) to estimate the conditional response distribution addressing the distributional changes across the predictor space, and (2) to identify important predictors for t ..."
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Cited by 33 (18 self)
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This article considers methodology for flexibly characterizing the relationship between a response and multiple predictors. Goals are (1) to estimate the conditional response distribution addressing the distributional changes across the predictor space, and (2) to identify important predictors for the response distribution change both with local regions and globally. We first introduce the probit stickbreaking process (PSBP) as a prior for an uncountable collection of predictordependent random probability measures and propose a PSBP mixture (PSBPM) of normal regressions for modeling the conditional distributions. A global variable selection structure is incorporated to discard unimportant predictors, while allowing estimation of posterior inclusion probabilities. Local variable selection is conducted relying on the conditional distribution estimates at different predictor points. An efficient stochastic search sampling algorithm is proposed for posterior computation. The methods are illustrated through simulation and applied to an epidemiologic study.
The discrete infinite logistic normal distribution
 Bayesian Analysis
"... We present the discrete infinite logistic normal distribution (DILN, “Dylan”), a Bayesian nonparametric prior for mixed membership models. DILN is a generalization of the hierarchical Dirichlet process (HDP) that models correlation structure between the weights of the atoms at the group level. We ..."
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Cited by 25 (7 self)
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We present the discrete infinite logistic normal distribution (DILN, “Dylan”), a Bayesian nonparametric prior for mixed membership models. DILN is a generalization of the hierarchical Dirichlet process (HDP) that models correlation structure between the weights of the atoms at the group level. We derive a representation of DILN as a normalized collection of gammadistributed random variables, and study its statistical properties. We consider applications to topic modeling and derive a variational Bayes algorithm for approximate posterior inference. We study the empirical performance of the DILN topic model on four corpora, comparing performance with the HDP and the correlated topic model. 1
A multivariate semiparametric Bayesian spatial modeling framework for hurricane surface wind fields
 Annals of Applied Statistics
, 2007
"... Storm surge, the onshore rush of sea water caused by the high winds and low pressure associated with a hurricane, can compound the effects of inland flooding caused by rainfall, leading to loss of property and loss of life for residents of coastal areas. Numerical ocean models are essential for crea ..."
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Cited by 19 (7 self)
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Storm surge, the onshore rush of sea water caused by the high winds and low pressure associated with a hurricane, can compound the effects of inland flooding caused by rainfall, leading to loss of property and loss of life for residents of coastal areas. Numerical ocean models are essential for creating storm surge forecasts for coastal areas. These models are driven primarily by the surface wind forcings. Currently, the gridded wind fields used by ocean models are specified by deterministic formulas that are based on the central pressure and location of the storm center. While these equations incorporate important physical knowledge about the structure of hurricane surface wind fields, they cannot always capture the asymmetric and dynamic nature of a hurricane. A new Bayesian multivariate spatial statistical modeling framework is introduced combining data with physical knowledge about the wind fields to improve the estimation of the wind vectors. Many spatial models assume the data follow a Gaussian distribution. However, this may be overlyrestrictive for wind fields data which often display erratic behavior, such as sudden changes in time or space. In this paper we develop a semiparametric multivariate spatial model for these data. Our model builds on the stickbreaking prior, which is frequently used in Bayesian modeling to capture uncertainty in the parametric form of an outcome. The stickbreaking prior is extended to the spatial setting by assigning each location a different, unknown distribution, and smoothing the distributions in space with a series of kernel functions. This semiparametric spatial model is shown to improve prediction compared to usual Bayesian Kriging methods for the wind field of Hurricane Ivan. 1. Introduction. Modeling
Hybrid Dirichlet mixture models for functional data
"... Summary. In functional data analysis, curves or surfaces are observed, up to measurement error, at a finite set of locations, for, say, a sample of n individuals. Often, the curves are homogeneous, except perhaps for individualspecific regions that provide heterogeneous behaviour (e.g. ‘damaged ’ a ..."
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Cited by 16 (0 self)
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Summary. In functional data analysis, curves or surfaces are observed, up to measurement error, at a finite set of locations, for, say, a sample of n individuals. Often, the curves are homogeneous, except perhaps for individualspecific regions that provide heterogeneous behaviour (e.g. ‘damaged ’ areas of irregular shape on an otherwise smooth surface). Motivated by applications with functional data of this nature, we propose a Bayesian mixture model, with the aim of dimension reduction, by representing the sample of n curves through a smaller set of canonical curves. We propose a novel prior on the space of probability measures for a random curve which extends the popular Dirichlet priors by allowing local clustering: nonhomogeneous portions of a curve can be allocated to different clusters and the n individual curves can be represented as recombinations (hybrids) of a few canonical curves. More precisely, the prior proposed envisions a conceptual hidden factor with klevels that acts locally on each curve. We discuss several models incorporating this prior and illustrate its performance with simulated and real data sets. We examine theoretical properties of the proposed finite hybrid Dirichlet mixtures, specifically, their behaviour as the number of the mixture components goes to 1 and their connection with Dirichlet process mixtures.
Copula Processes
"... We define a copula process which describes the dependencies between arbitrarily many random variables independently of their marginal distributions. As an example, we develop a stochastic volatility model, Gaussian Copula Process Volatility (GCPV), to predict the latent standard deviations of a sequ ..."
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Cited by 16 (4 self)
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We define a copula process which describes the dependencies between arbitrarily many random variables independently of their marginal distributions. As an example, we develop a stochastic volatility model, Gaussian Copula Process Volatility (GCPV), to predict the latent standard deviations of a sequence of random variables. To make predictions we use Bayesian inference, with the Laplace approximation, and with Markov chain Monte Carlo as an alternative. We find our model can outperform GARCH on simulated and financial data. And unlike GARCH, GCPV can easily handle missing data, incorporate covariates other than time, and model a rich class of covariance structures. Imagine measuring the distance of a rocket as it leaves Earth, and wanting to know how these measurements correlate with one another. How much does the value of the measurement at fifteen minutes depend on the measurement at five minutes? Once we’ve learned this correlation structure, suppose we want to compare it to the dependence between measurements of the rocket’s velocity. To do this, it is convenient to separate dependence from the marginal distributions of our measurements.
Dependent Indian Buffet Processes
"... Latent variable models represent hidden structure in observational data. To account for the distribution of the observational data changing over time, space or some other covariate, we need generalizations of latent variable models that explicitly capture this dependency on the covariate. A variety ..."
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Cited by 14 (4 self)
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Latent variable models represent hidden structure in observational data. To account for the distribution of the observational data changing over time, space or some other covariate, we need generalizations of latent variable models that explicitly capture this dependency on the covariate. A variety of such generalizations has been proposed for latent variable models based on the Dirichlet process. We address dependency on covariates in binary latent feature models, by introducing a dependent Indian buffet process. The model generates, for each value of the covariate, a binary random matrix with an unbounded number of columns. Evolution of the binary matrices over the covariate set is controlled by a hierarchical Gaussian process model. The choice of covariance functions controls the dependence structure and exchangeability properties of the model. We derive a Markov Chain Monte Carlo sampling algorithm for Bayesian inference, and provide experiments on both synthetic and realworld data. The experimental results show that explicit modeling of dependencies significantly improves accuracy of predictions. 1
Bayesian Nonparametric Modelling with the Dirichlet Process Regression Smoother
"... In this paper we discuss the problem of Bayesian fully nonparametric regression. A new construction of priors for nonparametric regression is discussed and a specific prior, the Dirichlet Process Regression Smoother, is proposed. We consider the problem of centring our process over a class of regres ..."
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Cited by 13 (0 self)
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In this paper we discuss the problem of Bayesian fully nonparametric regression. A new construction of priors for nonparametric regression is discussed and a specific prior, the Dirichlet Process Regression Smoother, is proposed. We consider the problem of centring our process over a class of regression models and propose fully nonparametric regression models with flexible location structures. Computational methods are developed for all models described. Results are presented for simulated and actual data examples. Keywords: Nonlinear regression; Nonparametric regression; Model centring; Stickbreaking prior