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14
Bayesian modeling of joint and conditional distributions. Unpublished manuscript
, 2009
"... In this paper, we study a Bayesian approach to flexible modeling of conditional distributions. The approach uses a flexible model for the joint distribution of the dependent and independent variables and then extracts the conditional distributions of interest from the estimated joint distribution. W ..."
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In this paper, we study a Bayesian approach to flexible modeling of conditional distributions. The approach uses a flexible model for the joint distribution of the dependent and independent variables and then extracts the conditional distributions of interest from the estimated joint distribution. We use a finite mixture of multivariate normals (FMMN) to estimate the joint distribution. The conditional distributions can then be assessed analytically or through simulations. The discrete variables are handled through the use of latent variables. The estimation procedure employs an MCMC algorithm. We provide a characterization of the Kullback–Leibler closure of FMMN and show that the joint and conditional predictive densities implied by FMMN model are consistent estimators for a large class of data generating processes with continuous and discrete observables. The method can be used as a robust regression model with discrete and continuous dependent and independent variables and as a Bayesian alternative to semi and nonparametric models such as quantile and kernel regression. In experiments, the method compares favorably with classical nonparametric and alternative Bayesian methods.
Autoregressive mixture models for dynamic spatial Poisson processes: Application to tracking intensity of violent crime
 Journal of the American Statistical Association
, 2010
"... This article develops a set of tools for smoothing and prediction with dependent point event patterns. The methodology is motivated by the problem of tracking weekly maps of violent crime events, but is designed to be straightforward to adapt to a wide variety of alternative settings. In particular, ..."
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This article develops a set of tools for smoothing and prediction with dependent point event patterns. The methodology is motivated by the problem of tracking weekly maps of violent crime events, but is designed to be straightforward to adapt to a wide variety of alternative settings. In particular, a Bayesian semiparametric framework is introduced for modeling correlated time series of marked spatial Poisson processes. The likelihood is factored into two independent components: the set of total integrated intensities and a series of process densities. For the former it is assumed that Poisson intensities are realizations from a dynamic linear model. In the latter case, a novel class of dependent stickbreaking mixture models are proposed to allow nonparametric density estimates to evolve in discrete time. This, a simple and flexible new model for dependent random distributions, is based on autoregressive time series of marginally beta random variables applied as correlated stickbreaking proportions. The approach allows for marginal Dirichlet process priors at each time and adds only a single new correlation term to the static model specification. Sequential Monte Carlo algorithms are described for online inference with each model component, and marginal likelihood calculations form the basis for inference about parameters governing temporal dynamics. Simulated examples are provided to illustrate the methodology, and we close with results for the motivating application of tracking violent crime in Cincinnati. M. A. Taddy is Assistant Professor of Econometrics and Statistics and Robert L. Graves Faculty
Simultaneous linear quantile regression: A semiparametric bayesian approach
 In press
, 2010
"... We introduce a semiparametric Bayesian framework for a simultaneous analysis of linear quantile regression models. A simultaneous analysis is essential to attain the true potential of the quantile regression framework, but is computationally challenging due to the associated monotonicity constraint ..."
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We introduce a semiparametric Bayesian framework for a simultaneous analysis of linear quantile regression models. A simultaneous analysis is essential to attain the true potential of the quantile regression framework, but is computationally challenging due to the associated monotonicity constraint on the quantile curves. For a univariate covariate, we present a simpler equivalent characterization of the monotonicity constraint through an interpolation of two monotone curves. The resulting formulation leads to a tractable likelihood function and is embedded within a Bayesian framework where the two monotone curves are modeled via logistic transformations of a smooth Gaussian process. A multivariate extension is proposed by combining the full support univariate model with a linear projection of the predictors. The resulting singleindex model remains easy to fit and provides substantial and measurable improvement over the first order linear heteroscedastic model. Two illustrative applications of the proposed method are provided.
Spatial quantile multiple regression using the asymmetric laplace process
 Bayesian Analysis
, 2012
"... Abstract. We consider quantile multiple regression through conditional quantile models, i.e. each quantile is modeled separately. We work in the context of spatially referenced data and extend the asymmetric Laplace model for quantile regression to a spatial process, the asymmetric Laplace process ( ..."
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Abstract. We consider quantile multiple regression through conditional quantile models, i.e. each quantile is modeled separately. We work in the context of spatially referenced data and extend the asymmetric Laplace model for quantile regression to a spatial process, the asymmetric Laplace process (ALP) for quantile regression with spatially dependent errors. By taking advantage of a convenient conditionally Gaussian representation of the asymmetric Laplace distribution, we are able to straightforwardly incorporate spatial dependence in this process. We develop the properties of this process under several specifications, each of which induces different smoothness and covariance behavior at the extreme quantiles. We demonstrate the advantages that may be gained by incorporating spatial dependence into this conditional quantile model by applying it to a data set of log selling prices of homes in Baton Rouge, LA, given characteristics of each house. We also introduce the asymmetric Laplace predictive process (ALPP) which accommodates large data sets, and apply it to a data set of birth weights given maternal covariates for several thousand births in North Carolina in 2000. By modeling the spatial structure in the data, we are able to show, using a check loss function, improved performance on each of the data sets for each of the quantiles at which the model was fit.
Particle Learning for General Mixtures
"... This paper develops efficient sequential learning methods for the estimation of general mixture models. The approach is distinguished from alternative particle filtering methods in two major ways. First, each iteration begins by resampling particles according to posterior predictive probability, lea ..."
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This paper develops efficient sequential learning methods for the estimation of general mixture models. The approach is distinguished from alternative particle filtering methods in two major ways. First, each iteration begins by resampling particles according to posterior predictive probability, leading to a more efficient set for propagation. Second, each particle tracks only the state of sufficient information for latent mixture components, thus leading to reduced dimensional inference. In addition, we describe how the approach will apply to more general mixture models of current interest in the literature; it is hoped that this will inspire a greater number of researchers to adopt sequential Monte Carlo methods for fitting their sophisticated mixture based models. Finally, we show that this particle learning approach leads to straightforward tools for marginal likelihood calculation and posterior cluster allocation. Specific versions of the algorithm are derived for standard density estimation applications based on both finite mixture models and Dirichlet process mixture models, as well as for the less common settings of latent feature selection through an Indian Buffet process and dependent distribution tracking through a probit stickbreaking model. Three simulation examples are presented: density estimation and model selection for a finite mixture model; a simulation study for Dirichlet process density estimation with as many as 12500 observations of 25 dimensional data, and an example of nonparametric mixture regression that requires learning truncated approximations to the infinite random mixing distribution.
Bayesian nonparametric mixture modeling for the intensity function of nonhomogeneous Poisson processes
 Department of
, 2005
"... (definitions, examples, posterior simulation methods) ..."
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(definitions, examples, posterior simulation methods)
A Fully Nonparametric Modelling Approach to Binary Regression
, 2014
"... We propose a general nonparametric Bayesian framework for binary regression, which is built from modelling for the joint responsecovariate distribution. The observed binary responses are assumed to arise from underlying continuous random variables through discretization, and we model the joint dist ..."
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We propose a general nonparametric Bayesian framework for binary regression, which is built from modelling for the joint responsecovariate distribution. The observed binary responses are assumed to arise from underlying continuous random variables through discretization, and we model the joint distribution of these latent responses and the covariates using a Dirichlet process mixture of multivariate normals. We show that the kernel of the induced mixture model for the observed data is identifiable upon a restriction on the latent variables. To allow for appropriate dependence structure while facilitating identifiability, we use a squarerootfree Cholesky decomposition of the covariance matrix in the normal mixture kernel. In addition to allowing for the necessary restriction, this modelling strategy provides substantial simplifications in implementation of Markov chain Monte Carlo posterior simulation. We illustrate the utility of the modelling approach with two data examples, and discuss extensions to incorporate multivariate ordinal responses, as well as mixed ordinalcontinuous responses.
Expert Information and Nonparametric Bayesian Inference of Rare Events
, 2012
"... Prior distributions are important in Bayesian inference of rare events because historical data information is scarce, and experts are an important source of information for elicitation of a prior distribution. We propose a method to incorporate expert information into nonparametric Bayesian inferenc ..."
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Prior distributions are important in Bayesian inference of rare events because historical data information is scarce, and experts are an important source of information for elicitation of a prior distribution. We propose a method to incorporate expert information into nonparametric Bayesian inference on rare events when expert knowledge is elicited as moment conditions on a finite dimensional parameter θ only. We generalize the Dirichlet process mixture model to merge expert information into the Dirichlet process (DP) prior to satisfy expert’s moment conditions. Among all the priors that comply with expert knowledge, we use the one that is closest to the original DP prior in the KullbackLeibler information criterion. The resulting prior distribution is given by exponentially tilting the DP prior along θ. We provide a MetropolisHastings algorithm to implement our approach to sample from posterior distributions with exponentially tilted DP priors. Our method combines prior information from an econometrician and an expert by finding the leastinformative prior given expert information.
Markov Switching Dirichlet Process Mixture Regression
, 2009
"... Markov switching models can be used to study heterogeneous populations that are observed over time. This paper explores modeling the group characteristics nonparametrically, under both homogeneous and nonhomogeneous Markov switching for group probabilities. The model formulation involves a finite m ..."
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Markov switching models can be used to study heterogeneous populations that are observed over time. This paper explores modeling the group characteristics nonparametrically, under both homogeneous and nonhomogeneous Markov switching for group probabilities. The model formulation involves a finite mixture of conditionally independent Dirichlet process mixtures, with a Markov chain defining the mixing distribution. The proposed methodology focuses on settings where the number of subpopulations is small and can be assumed to be known, and flexible modeling is required for group regressions. We develop Dirichlet process mixture prior probability models for the joint distribution of individual group responses and covariates. The implied conditional distribution of the response given the covariates is then used for inference. The modeling framework allows for both nonlinearities in the resulting regression functions and nonstandard shapes in the response distributions. We design a simulationbased model fitting method for full posterior inference. Furthermore, we propose a general approach for inclusion of external covariates dependent on the Markov chain but conditionally independent from the response. The methodology is applied to a problem from fisheries research involving analysis of stockrecruitment data under shifts in the ecosystem state.