arXiv subjects

David B. Dunson

Publications and source records attributed to David B. Dunson.

At least 181 records · Page 10Linked to original sources

arXiv2013

Bayesian Compressed Regression

As an alternative to variable selection or shrinkage in high dimensional regression, we propose to randomly compress the predictors prior to analysis. This dramatically reduces storage and computational bottlenecks, performing well when the predictors can be projected to a low dimensional linear subspace with minimal loss of information about the response. As opposed to existing Bayesian dimensionality reduction approaches, the exact posterior distribution conditional on the compressed data is available analytically, speeding up computation by many orders of magnitude while also bypassing robustness issues due to convergence and mixing problems with MCMC. Model averaging is used to reduce sensitivity to the random projection matrix, while accommodating uncertainty in the subspace dimension. Strong theoretical support is provided for the approach by showing near parametric convergence rates for the predictive density in the large p small n asymptotic paradigm. Practical performance relative to competitors is illustrated in simulations and real data applications.

stat.ML↗

arXiv2013

Bayesian learning of joint distributions of objects

There is increasing interest in broad application areas in defining flexible joint models for data having a variety of measurement scales, while also allowing data of complex types, such as functions, images and documents. We consider a general framework for nonparametric Bayes joint modeling through mixture models that incorporate dependence across data types through a joint mixing measure. The mixing measure is assigned a novel infinite tensor factorization (ITF) prior that allows flexible dependence in cluster allocation across data types. The ITF prior is formulated as a tensor product of stick-breaking processes. Focusing on a convenient special case corresponding to a Parafac factorization, we provide basic theory justifying the flexibility of the proposed prior and resulting asymptotic properties. Focusing on ITF mixtures of product kernels, we develop a new Gibbs sampling algorithm for routine implementation relying on slice sampling. The methods are compared with alternative joint mixture models based on Dirichlet processes and related approaches through simulations and real data applications.

stat.ME↗

arXiv2013

Bayesian Consensus Clustering

The task of clustering a set of objects based on multiple sources of data arises in several modern applications. We propose an integrative statistical model that permits a separate clustering of the objects for each data source. These separate clusterings adhere loosely to an overall consensus clustering, and hence they are not independent. We describe a computationally scalable Bayesian framework for simultaneous estimation of both the consensus clustering and the source-specific clusterings. We demonstrate that this flexible approach is more robust than joint clustering of all data sources, and is more powerful than clustering each data source separately. This work is motivated by the integrated analysis of heterogeneous biomedical data, and we present an application to subtype identification of breast cancer tumor samples using publicly available data from The Cancer Genome Atlas. Software is available at http://people.duke.edu/~el113/software.html.

stat.ML↗

arXiv2013

Bayesian Conditional Tensor Factorizations for High-Dimensional Classification

In many application areas, data are collected on a categorical response and high-dimensional categorical predictors, with the goals being to build a parsimonious model for classification while doing inferences on the important predictors. In settings such as genomics, there can be complex interactions among the predictors. By using a carefully-structured Tucker factorization, we define a model that can characterize any conditional probability, while facilitating variable selection and modeling of higher-order interactions. Following a Bayesian approach, we propose a Markov chain Monte Carlo algorithm for posterior computation accommodating uncertainty in the predictors to be included. Under near sparsity assumptions, the posterior distribution for the conditional probability is shown to achieve close to the parametric rate of contraction even in ultra high-dimensional settings. The methods are illustrated using simulation examples and biomedical applications.

stat.ME↗

arXiv2013

Bayesian Gaussian Copula Factor Models for Mixed Data

Gaussian factor models have proven widely useful for parsimoniously characterizing dependence in multivariate data. There is a rich literature on their extension to mixed categorical and continuous variables, using latent Gaussian variables or through generalized latent trait models acommodating measurements in the exponential family. However, when generalizing to non-Gaussian measured variables the latent variables typically influence both the dependence structure and the form of the marginal distributions, complicating interpretation and introducing artifacts. To address this problem we propose a novel class of Bayesian Gaussian copula factor models which decouple the latent factors from the marginal distributions. A semiparametric specification for the marginals based on the extended rank likelihood yields straightforward implementation and substantial computational gains, critical for scaling to high-dimensional applications. We provide new theoretical and empirical justifications for using this likelihood in Bayesian inference. We propose new default priors for the factor loadings and develop efficient parameter-expanded Gibbs sampling for posterior computation. The methods are evaluated through simulations and applied to a dataset in political science. The methods in this paper are implemented in the R package bfa.

stat.ME↗

arXiv2012

Bayesian shrinkage

Penalized regression methods, such as $L_1$ regularization, are routinely used in high-dimensional applications, and there is a rich literature on optimality properties under sparsity assumptions. In the Bayesian paradigm, sparsity is routinely induced through two-component mixture priors having a probability mass at zero, but such priors encounter daunting computational problems in high dimensions. This has motivated an amazing variety of continuous shrinkage priors, which can be expressed as global-local scale mixtures of Gaussians, facilitating computation. In sharp contrast to the corresponding frequentist literature, very little is known about the properties of such priors. Focusing on a broad class of shrinkage priors, we provide precise results on prior and posterior concentration. Interestingly, we demonstrate that most commonly used shrinkage priors, including the Bayesian Lasso, are suboptimal in high-dimensional settings. A new class of Dirichlet Laplace (DL) priors are proposed, which are optimal and lead to efficient posterior computation exploiting results from normalized random measure theory. Finite sample performance of Dirichlet Laplace priors relative to alternatives is assessed in simulations.

math.ST↗

arXiv2012

Stochastic Volatility Regression for Functional Data Dynamics

Although there are many methods for functional data analysis (FDA), little emphasis is put on characterizing variability among volatilities of individual functions. In particular, certain individuals exhibit erratic swings in their trajectory while other individuals have more stable trajectories. There is evidence of such volatility heterogeneity in blood pressure trajectories during pregnancy, for example, and reason to suspect that volatility is a biologically important feature. Most FDA models implicitly assume similar or identical smoothness of the individual functions, and hence can lead to misleading inferences on volatility and an inadequate representation of the functions. We propose a novel class of FDA models characterized using hierarchical stochastic differential equations. We model the derivatives of a mean function and deviation functions using Gaussian processes, while also allowing covariate dependence including on the volatilities of the deviation functions. Following a Bayesian approach to inference, a Markov chain Monte Carlo algorithm is used for posterior computation. The methods are tested on simulated data and applied to blood pressure trajectories during pregnancy.

stat.AP↗

arXiv2012

Repulsive Mixtures

Discrete mixture models are routinely used for density estimation and clustering. While conducting inferences on the cluster-specific parameters, current frequentist and Bayesian methods often encounter problems when clusters are placed too close together to be scientifically meaningful. Current Bayesian practice generates component-specific parameters independently from a common prior, which tends to favor similar components and often leads to substantial probability assigned to redundant components that are not needed to fit the data. As an alternative, we propose to generate components from a repulsive process, which leads to fewer, better separated and more interpretable clusters. We characterize this repulsive prior theoretically and propose a Markov chain Monte Carlo sampling algorithm for posterior computation. The methods are illustrated using simulated data as well as real datasets.

stat.ME↗

arXiv2012

Multiresolution Gaussian Processes

We propose a multiresolution Gaussian process to capture long-range, non-Markovian dependencies while allowing for abrupt changes. The multiresolution GP hierarchically couples a collection of smooth GPs, each defined over an element of a random nested partition. Long-range dependencies are captured by the top-level GP while the partition points define the abrupt changes. Due to the inherent conjugacy of the GPs, one can analytically marginalize the GPs and compute the conditional likelihood of the observations given the partition tree. This property allows for efficient inference of the partition itself, for which we employ graph-theoretic techniques. We apply the multiresolution GP to the analysis of Magnetoencephalography (MEG) recordings of brain activity.

stat.ME↗

arXiv2012

Bayesian modeling of temporal dependence in large sparse contingency tables

In many applications, it is of interest to study trends over time in relationships among categorical variables, such as age group, ethnicity, religious affiliation, political party and preference for particular policies. At each time point, a sample of individuals provide responses to a set of questions, with different individuals sampled at each time. In such settings, there tends to be abundant missing data and the variables being measured may change over time. At each time point, one obtains a large sparse contingency table, with the number of cells often much larger than the number of individuals being surveyed. To borrow information across time in modeling large sparse contingency tables, we propose a Bayesian autoregressive tensor factorization approach. The proposed model relies on a probabilistic Parafac factorization of the joint pmf characterizing the categorical data distribution at each time point, with autocorrelation included across times. Efficient computational methods are developed relying on MCMC. The methods are evaluated through simulation examples and applied to social survey data.

stat.ME↗

arXiv2012

Marginally Specified Priors for Nonparametric Bayesian Estimation

Prior specification for nonparametric Bayesian inference involves the difficult task of quantifying prior knowledge about a parameter of high, often infinite, dimension. Realistically, a statistician is unlikely to have informed opinions about all aspects of such a parameter, but may have real information about functionals of the parameter, such the population mean or variance. This article proposes a new framework for nonparametric Bayes inference in which the prior distribution for a possibly infinite-dimensional parameter is decomposed into two parts: an informative prior on a finite set of functionals, and a nonparametric conditional prior for the parameter given the functionals. Such priors can be easily constructed from standard nonparametric prior distributions in common use, and inherit the large support of the standard priors upon which they are based. Additionally, posterior approximations under these informative priors can generally be made via minor adjustments to existing Markov chain approximation algorithms for standard nonparametric prior distributions. We illustrate the use of such priors in the context of multivariate density estimation using Dirichlet process mixture models, and in the modeling of high-dimensional sparse contingency tables.

stat.ME↗

arXiv2012

Generalized Beta Mixtures of Gaussians

In recent years, a rich variety of shrinkage priors have been proposed that have great promise in addressing massive regression problems. In general, these new priors can be expressed as scale mixtures of normals, but have more complex forms and better properties than traditional Cauchy and double exponential priors. We first propose a new class of normal scale mixtures through a novel generalized beta distribution that encompasses many interesting priors as special cases. This encompassing framework should prove useful in comparing competing priors, considering properties and revealing close connections. We then develop a class of variational Bayes approximations through the new hierarchy presented that will scale more efficiently to the types of truly massive data sets that are now encountered routinely.

stat.ME↗

arXiv2012

Locally Adaptive Bayes Nonparametric Regression via Nested Gaussian Processes

We propose a nested Gaussian process (nGP) as a locally adaptive prior for Bayesian nonparametric regression. Specified through a set of stochastic differential equations (SDEs), the nGP imposes a Gaussian process prior for the function's $m$th-order derivative. The nesting comes in through including a local instantaneous mean function, which is drawn from another Gaussian process inducing adaptivity to locally-varying smoothness. We discuss the support of the nGP prior in terms of the closure of a reproducing kernel Hilbert space, and consider theoretical properties of the posterior. The posterior mean under the nGP prior is shown to be equivalent to the minimizer of a nested penalized sum-of-squares involving penalties for both the global and local roughness of the function. Using highly-efficient Markov chain Monte Carlo for posterior inference, the proposed method performs well in simulation studies compared to several alternatives, and is scalable to massive data, illustrated through a proteomics application.

stat.ME↗

arXiv2012

Path Following and Empirical Bayes Model Selection for Sparse Regression

In recent years, a rich variety of regularization procedures have been proposed for high dimensional regression problems. However, tuning parameter choice and computational efficiency in ultra-high dimensional problems remain vexing issues. The routine use of $\ell_1$ regularization is largely attributable to the computational efficiency of the LARS algorithm, but similar efficiency for better behaved penalties has remained elusive. In this article, we propose a highly efficient path following procedure for combination of any convex loss function and a broad class of penalties. From a Bayesian perspective, this algorithm rapidly yields maximum a posteriori estimates at different hyper-parameter values. To bypass the inefficiency and potential instability of cross validation, we propose an empirical Bayes procedure for rapidly choosing the optimal model and corresponding hyper-parameter value. This approach applies to any penalty that corresponds to a proper prior distribution on the regression coefficients. While we mainly focus on sparse estimation of generalized linear models, the method extends to more general regularizations such as polynomial trend filtering after reparameterization. The proposed algorithm scales efficiently to large $p$ and/or $n$. Solution paths of 10,000 dimensional examples are computed within one minute on a laptop for various generalized linear models (GLM). Operating characteristics are assessed through simulation studies and the methods are applied to several real data sets.

stat.CO↗

arXiv2012

Bayesian hierarchical modeling of simply connected 2D shapes

Models for distributions of shapes contained within images can be widely used in biomedical applications ranging from tumor tracking for targeted radiation therapy to classifying cells in a blood sample. Our focus is on hierarchical probability models for the shape and size of simply connected 2D closed curves, avoiding the need to specify landmarks through modeling the entire curve while borrowing information across curves for related objects. Prevalent approaches follow a fundamentally different strategy in providing an initial point estimate of the curve and/or locations of landmarks, which are then fed into subsequent statistical analyses. Such two-stage methods ignore uncertainty in the first stage, and do not allow borrowing of information across objects in estimating object shapes and sizes. Our fully Bayesian hierarchical model is based on multiscale deformations within a linear combination of cyclic basis characterization, which facilitates automatic alignment of the different curves accounting for uncertainty. The characterization is shown to be highly flexible in representing 2D closed curves, leading to a nonparametric Bayesian prior with large support. Efficient Markov chain Monte Carlo methods are developed for simultaneous analysis of many objects. The methods are evaluated through simulation examples and applied to yeast cell imaging data.

stat.ME↗

arXiv2011

Generalized Admixture Mapping for Complex Traits

Admixture mapping is a popular tool to identify regions of the genome associated with traits in a recently admixed population. Existing methods have been developed primarily for identification of a single locus influencing a dichotomous trait within a case-control study design. We propose a generalized admixture mapping (GLEAM) approach, a flexible and powerful regression method for both quantitative and qualitative traits, which is able to test for association between the trait and local ancestries in multiple loci simultaneously and adjust for covariates. The new method is based on the generalized linear model and utilizes a quadratic normal moment prior to incorporate admixture prior information. Through simulation, we demonstrate that GLEAM achieves lower type I error rate and higher power than existing methods both for qualitative traits and more significantly for quantitative traits. We applied GLEAM to genome-wide SNP data from the Illumina African American panel derived from a cohort of black woman participating in the Healthy Pregnancy, Healthy Baby study and identified a locus on chromosome 2 associated with the averaged maternal mean arterial pressure during 24 to 28 weeks of pregnancy.

stat.AP↗

arXiv2011

Adverse Subpopulation Regression for Multivariate Outcomes with High-Dimensional Predictors

Biomedical studies have a common interest in assessing relationships between multiple related health outcomes and high-dimensional predictors. For example, in reproductive epidemiology, one may collect pregnancy outcomes such as length of gestation and birth weight and predictors such as single nucleotide polymorphisms in multiple candidate genes and environmental exposures. In such settings, there is a need for simple yet flexible methods for selecting true predictors of adverse health responses from a high-dimensional set of candidate predictors. To address this problem, one may either consider linear regression models for the continuous outcomes or convert these outcomes into binary indicators of adverse responses using pre-defined cutoffs. The former strategy has the disadvantage of often leading to a poorly fitting model that does not predict risk well, while the latter approach can be very sensitive to the cutoff choice. As a simple yet flexible alternative, we propose a method for adverse subpopulation regression (ASPR), which relies on a two component latent class model, with the dominant component corresponding to (presumed) healthy individuals and the risk of falling in the minority component characterized via a logistic regression. The logistic regression model is designed to accommodate high-dimensional predictors, as occur in studies with a large number of gene by environment interactions, through use of a flexible nonparametric multiple shrinkage approach. The Gibbs sampler is developed for posterior computation. The methods are evaluated using simulation studies and applied to a genetic epidemiology study of pregnancy outcomes.

stat.AP↗

arXiv2011

Multivariate convex regression with adaptive partitioning

We propose a new, nonparametric method for multivariate regression subject to convexity or concavity constraints on the response function. Convexity constraints are common in economics, statistics, operations research, financial engineering and optimization, but there is currently no multivariate method that is computationally feasible for more than a few hundred observations. We introduce Convex Adaptive Partitioning (CAP), which creates a globally convex regression model from locally linear estimates fit on adaptively selected covariate partitions. CAP is computationally efficient, in stark contrast to current methods. The most popular method, the least squares estimator, has a computational complexity of $\mathcal{O}(n^3)$. We show that CAP has a computational complexity of $\mathcal{O}(n \log(n)\log(\log(n)))$ and also give consistency results. CAP is applied to value function approximation for pricing American basket options with a large number of underlying assets.

stat.ME↗