DOCSLIB.ORG

Rank-Based Tempo-Spatial Clustering: a Framework for Rapid Outbreak Detection Using Single Or Multiple Data Streams

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Rank-Based Tempo-Spatial Clustering: a Framework for Rapid Outbreak Detection Using Single Or Multiple Data Streams RANK-BASED TEMPO-SPATIAL CLUSTERING: A FRAMEWORK FOR RAPID OUTBREAK DETECTION USING SINGLE OR MULTIPLE DATA STREAMS by Jialan Que Bachelor of Science, Nankai University, China, 2001 Master of Science, Nankai University, China, 2004 Master of Science, University of Pittsburgh, 2008 Submitted to the Graduate Faculty of Intelligent Systems Program in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 2012 UNIVERSITY OF PITTSBURGH INTELLIGENT SYSTEMS PROGRAM This dissertation was presented by Jialan Que It was defended on April 13, 2012 and approved by Gregory F. Cooper, Associate Professor, Biomedical Informatics and Intelligent Systems Roger S. Day, Associate Professor, Biomedical Informatics and Biostatistics Milos Hauskrecht, Associate Professor, Computer Science Dissertation Advisor: Fu-Chiang Tsui, Research Assistant Professor, Biomedical Informatics and Intelligent Systems ii Copyright © by Jialan Que 2012 iii RANK-BASED TEMPO-SPATIAL CLUSTERING: A FRAMEWORK FOR RAPID OUTBREAK DETECTION USING SINGLE OR MULTIPLE DATA STREAMS Jialan Que, PhD University of Pittsburgh, 2012 In recent decades, algorithms for disease outbreak detection have become one of the main interests of public health practitioners as a way to identify and localize an outbreak as early as possible in order to inform further public health response to prevent a pandemic from developing. Today’s increased threat of biological warfare and terrorism provide an even stronger impetus to develop methods for outbreak detection based on symptoms as well as definitive laboratory diagnoses. In this dissertation work, I explore the problems inherent to rapid disease outbreak detection using both spatial and temporal information. I develop a framework of non- parameterized algorithms which search for patterns of disease outbreak in spatial sub- regions of the monitored region within a certain period. Compared to the current existing spatial or tempo-spatial algorithm, the algorithms in this framework provide a methodology for fast searching of either a univariate data set or multivariate data set. It first measures how likely a study area has an outbreak occur given the baseline data and currently observed data. Then it applies a greedy searching mechanism to look for clusters with high posterior probabilities given the risk measurement for each unit area as a heuristic. The performance of the proposed algorithms is then evaluated. From the perspective of predictive modeling, I adopted a Gamma-Poisson (GP) model to compute the probability of having an outbreak in each cluster when analyzing iv univariate data. I built a multinomial generalized Dirichlet (MGD) model to identify outbreak clusters from multivariate data that include the OTC data streams collected by the national retail data monitor (NRDM) [1] and the ED data streams collected by the RODS system [2]. Key contributions of this dissertation include 1) the introduction of a rank-based tempo-spatial clustering algorithm, RSC, which utilizes greedy searching and a Bayesian GP model for disease outbreak detection with comparable detection timeliness, cluster positive prediction value (PPV) and improved running time; 2) the proposing of a multivariate extension of RSC (MRSC) which applies an MGD model. The evaluation demonstrates the advantage of the MGD model in effectively suppressing the false alarms caused by baseline shifts. v TABLE OF CONTENTS TABLE OF CONTENTS ................................................................................................... vi LIST OF TABLES ............................................................................................................. ix LIST OF FIGURES ........................................................................................................... xi 1.0 Introduction .............................................................................................................. 1 1.1 Research domain .................................................................................................. 2 1.2 Overview of the proposed methodology .............................................................. 4 1.3 Dissertation hypothesis ........................................................................................ 5 1.4 Guide for the reader.............................................................................................. 6 2.0 Background .............................................................................................................. 9 2.1 Bayesian framework ........................................................................................... 10 2.1.1 Bayes’ Theorem ...................................................................................... 10 2.2 Priors .................................................................................................................. 10 2.3 Several statistical distributions ........................................................................... 12 2.3.1 Gamma and Poisson distribution ............................................................ 12 2.3.2 Multinomial distribution ......................................................................... 12 2.3.3 Dirichlet distribution ............................................................................... 14 2.3.4 Generalized Dirichlet distribution ........................................................... 17 3.0 Related work .......................................................................................................... 20 3.1 Temporal detection methods .............................................................................. 21 3.2 Spatial and tempo-spatial detection methods ..................................................... 21 3.2.1 Frequentist approaches ............................................................................ 22 3.2.2 Bayesian approaches ............................................................................... 24 3.2.3 Issues of algorithms using scan statistics ................................................ 26 3.3 Multivariate methods.......................................................................................... 28 3.4 Calculation of baselines ..................................................................................... 35 3.5 Computational considerations ............................................................................ 37 vi 3.6 The hypothetical advantages of proposed algorithms ........................................ 38 4.0 The experimental domain ...................................................................................... 41 4.1 Background data ................................................................................................. 42 4.1.1 ED data .................................................................................................... 42 4.1.2 OTC data ................................................................................................. 43 4.1.3 OTC data simulation model using NRDM data ...................................... 43 4.2 Linear outbreak simulator .................................................................................. 49 4.3 Multivariate spatial-temporal event simulator ................................................... 50 4.3.1 Parameters ............................................................................................... 51 4.3.2 The model ................................................................................................ 52 5.0 Rank-based tempo-spatial clustering (RSC) .......................................................... 55 5.1 An algorithm for early outbreak detection—rank-based tempo-spatial clustering (RSC) ............................................................................................................................ 55 5.1.1 Risk rate assessment ................................................................................ 56 5.1.2 Adjacency criterion ................................................................................. 59 5.1.3 Searching for clusters .............................................................................. 60 5.1.4 Priors ....................................................................................................... 61 5.1.5 Computing posterior probability of a cluster .......................................... 62 5.1.6 The temporal window.............................................................................. 63 5.1.7 Experiments ............................................................................................. 64 5.1.8 Discussion ............................................................................................... 72 5.2 Grid-based RSC (G-RSC) .................................................................................. 74 5.2.1 Experiments ............................................................................................. 75 5.2.2 Experimental results ................................................................................ 76 5.2.3 Discussion ............................................................................................... 77 6.0 The multivariate rank-based clustering (MRSC) ................................................... 80 6.1 Computing likelihoods using the Multinomial-generalized-Dirichlet (MGD) model ............................................................................................................................ 84 6.2 Bayesian
Load more

Recommended publications
  • Efficient Empirical Bayes Variable Selection and Estimation in Linear
    Efficient Empirical Bayes Variable Selection and Estimation in Linear Models Ming YUAN and Yi LIN We propose an empirical Bayes method for variable selection and coefficient estimation in linear regression models. The method is based on a particular hierarchical Bayes formulation, and the empirical Bayes estimator is shown to be closely related to the LASSO estimator. Such a connection allows us to take advantage of the recently developed quick LASSO algorithm to compute the empirical Bayes estimate, and provides a new way to select the tuning parameter in the LASSO method. Unlike previous empirical Bayes variable selection methods, which in most practical situations can be implemented only through a greedy stepwise algorithm, our method gives a global solution efficiently. Simulations and real examples show that the proposed method is very competitive in terms of variable selection, estimation accuracy, and computation speed compared with other variable selection and estimation methods. KEY WORDS: Hierarchical model; LARS algorithm; LASSO; Model selection; Penalized least squares. 1. INTRODUCTION because the number of candidate models grows exponentially as the number of predictors increases. In practice, this type of We consider the problem of variable selection and coeffi- method is implemented in a stepwise fashion through forward cient estimation in the common normal linear regression model selection or backward elimination. In doing so, one contents wherewehaven observations on a dependent variable Y and oneself with the locally optimal solution instead of the globally p predictors (x , x ,...,x ), and 1 2 p optimal solution. Y = Xβ + , (1) A number of other variable selection methods have been in- 2 troduced in recent years (George and McCulloch 1993; Foster where ∼ Nn(0,σ I) and β = (β1,...,βp) .
    [Show full text]
  • Bayes and Empirical-Bayes Multiplicity Adjustment in the Variable-Selection Problem
    The Annals of Statistics 2010, Vol. 38, No. 5, 2587–2619 DOI: 10.1214/10-AOS792 c Institute of Mathematical Statistics, 2010 BAYES AND EMPIRICAL-BAYES MULTIPLICITY ADJUSTMENT IN THE VARIABLE-SELECTION PROBLEM By James G. Scott1 and James O. Berger University of Texas at Austin and Duke University This paper studies the multiplicity-correction effect of standard Bayesian variable-selection priors in linear regression. Our first goal is to clarify when, and how, multiplicity correction happens auto- matically in Bayesian analysis, and to distinguish this correction from the Bayesian Ockham’s-razor effect. Our second goal is to con- trast empirical-Bayes and fully Bayesian approaches to variable selec- tion through examples, theoretical results and simulations. Consider- able differences between the two approaches are found. In particular, we prove a theorem that characterizes a surprising aymptotic dis- crepancy between fully Bayes and empirical Bayes. This discrepancy arises from a different source than the failure to account for hyper- parameter uncertainty in the empirical-Bayes estimate. Indeed, even at the extreme, when the empirical-Bayes estimate converges asymp- totically to the true variable-inclusion probability, the potential for a serious difference remains. 1. Introduction. This paper addresses concerns about multiplicity in the traditional variable-selection problem for linear models. We focus on Bayesian and empirical-Bayesian approaches to the problem. These methods both have the attractive feature that they can, if set up correctly, account for multiplicity automatically, without the need for ad-hoc penalties. Given the huge number of possible predictors in many of today’s scien- tific problems, these concerns about multiplicity are becoming ever more relevant.
    [Show full text]
  • Approximate Empirical Bayes for Deep Neural Networks
    Approximate Empirical Bayes for Deep Neural Networks Han Zhao,∗ Yao-Hung Hubert Tsai,∗ Ruslan Salakhutdinov, Geoff Gordon {hanzhao, yaohungt, rsalakhu, ggordon}@cs.cmu.edu Machine Learning Department Carnegie Mellon University Abstract Dropout [22] and the more recent DeCov [5] method, in this paper we propose a regularization approach for the We propose an approximate empirical Bayes weight matrix in neural networks through the lens of the framework and an efficient algorithm for learn- empirical Bayes method. We aim to address the prob- ing the weight matrix of deep neural networks. lem of overfitting when training large networks on small Empirically, we show the proposed method dataset. Our key insight stems from the famous argument works as a regularization approach that helps by Efron [6]: It is beneficial to learn from the experience generalization when training neural networks of others. Specifically, from an algorithmic perspective, on small datasets. we argue that the connection weights of neurons in the same layer (row/column vectors of the weight matrix) will be correlated with each other through the backpropaga- 1 Introduction tion learning, and hence by learning from other neurons in the same layer, a neuron can “borrow statistical strength” The empirical Bayes methods provide us a powerful tool from other neurons to reduce overfitting. to obtain Bayesian estimators even if we do not have As an illustrating example, consider the simple setting complete information about the prior distribution. The where the input x 2 Rd is fully connected to a hidden literature on the empirical Bayes methods and its appli- layer h 2 Rp, which is further fully connected to the cations is abundant [2, 6, 8, 9, 10, 15, 21, 23].
    [Show full text]
  • Empirical Bayes Least Squares Estimation Without an Explicit Prior
    Computer Science Technical Report TR2007-900, May 2007 Courant Institute of Mathematical Sciences, New York University http://cs.nyu.edu/web/Research/TechReports/reports.html Empirical Bayes Least Squares Estimation without an Explicit Prior Martin Raphan Eero P. Simoncelli Howard Hughes Medical Institute, Courant Inst. of Mathematical Sciences Center for Neural Science, and New York University Courant Inst. of Mathematical Sciences [email protected] New York University [email protected] Bayesian estimators are commonly constructed using an explicit prior model. In many applications, one does not have such a model, and it is difficult to learn since one does not have access to uncorrupted measure- ments of the variable being estimated. In many cases however, including the case of contamination with additive Gaussian noise, the Bayesian least squares estimator can be formulated directly in terms of the distri- bution of noisy measurements. We demonstrate the use of this formu- lation in removing noise from photographic images. We use a local ap- proximation of the noisy measurement distribution by exponentials over adaptively chosen intervals, and derive an estimator from this approxi- mate distribution. We demonstrate through simulations that this adaptive Bayesian estimator performs as well or better than previously published estimators based on simple prior models. 1 Introduction Denoising is a classic signal processing problem, and Bayesian methods can provide well formulated and highly successful solutions. Bayesian estimators are derived from three fundamental components: 1) the likelihood function, which characterizes the distortion process, 2) the prior, which characterizes the distribution of the variable to be estimated in the absence of any measurement data, and 3) the loss function, which expresses the cost of making errors.
    [Show full text]
  • Nonparametric Empirical Bayes for the Dirichlet Process Mixture Model
    Nonparametric empirical Bayes for the Dirichlet process mixture model Jon D. McAuliffe David M. Blei Michael I. Jordan October 4, 2004 Abstract 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 infinite-dimensional 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 α, the other Dirichlet process parameter, that is inspired by an existing characterization of its maximum-likelihood 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 real-world datasets illustrates the robustness of this approach. 1 INTRODUCTION Mixture modeling is an effective and widely practiced density estimation method, capable of representing the phe- nomena that underlie many real-world datasets. However, a long-standing difficulty in mixture analysis is choosing the number of mixture components. Model selection methods, such as cross-validation or methods based on Bayes factors, treat the number of components as an unknown constant and set its value based on the observed data.
    [Show full text]
  • The Empirical Bayes Estimation of an Instantaneous Spike Rate with a Gaussian Process Prior
    The empirical Bayes estimation of an instantaneous spike rate with a Gaussian process prior Shinsuke Koyama Department of Physics, Kyoto University Department of Statistics and Center for Neural Basis of Cognition Carnegie Mellon University Pittsburgh, PA 15213 [email protected] Takeaki Shimokawa Department of Physics, Kyoto University Sakyo-ku, Kyoto 606-8502, Japan [email protected] Shigeru Shinomoto Department of Physics, Kyoto University Sakyo-ku, Kyoto 606-8502, Japan [email protected] Abstract We investigate a Bayesian method for systematically capturing the underlying fir- ing rate of a neuron. Any method of rate estimation requires a prior assumption about the flatness of the underlying rate, which can be represented by a Gaussian process prior. This Bayesian framework enables us to adjust the very assump- tion by taking into account the distribution of raw data: A hyperparameter of the Gaussian process prior is selected so that the marginal likelihood is maximized. It takes place that this hyperparameter diverges for spike sequences derived from a moderately fluctuating rate. By utilizing the path integral method, we demonstrate two cases that exhibit the divergence continuously and discontinuously. 1 Introduction Bayesian methods based on Gaussian process priors have recently become quite popular, and provide flexible non-parametric methods to machine learning tasks such as regression and classification problems [1, 2, 3]. The important advantage of Gaussian process models is the explicit probabilistic formulation, which not only provides probabilistic predictions but also gives the ability to infer hyperparameters of models [4]. One of the interesting applications of the Gaussian process is the analysis of neuronal spike se- quences.
    [Show full text]
  • Empirical Bayes Posterior Concentration in Sparse High
    Empirical Bayes posterior concentration in sparse high-dimensional linear models Ryan Martin∗ Raymond Mess Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago (rgmartin, rmess1)@uic.edu Stephen G. Walker Department of Mathematics University of Texas at Austin [email protected] December 6, 2018 Abstract We propose a new empirical Bayes approach for inference in the p ≫ n normal linear model. The novelty is the use of data in the prior in two ways, for centering and regularization. Under suitable sparsity assumptions, we establish a variety of concentration rate results for the empirical Bayes posterior distribution, relevant for both estimation and model selection. Computation is straightforward and fast, and simulation results demonstrate the strong finite-sample performance of the empirical Bayes model selection procedure. Keywords and phrases: Data-dependent prior; fractional likelihood; minimax rate; regression; variable selection. arXiv:1406.7718v5 [math.ST] 5 Dec 2018 1 Introduction In this paper, we consider the Gaussian linear regression model, given by Y = Xβ + ε, (1) where Y is a n × 1 vector of response variables, X is a n × p matrix of predictor variables, β is a p × 1 vector of slope coefficients, and ε is a n × 1 vector of iid N(0, σ2) random errors. Recently, there has been considerable interest in the high-dimensional case, where p ≫ n, driven primarily by challenging applications. Indeed, in genetic studies, where ∗As of August 2016, RM is affiliated with North Carolina State University, [email protected]. 1 the response variable corresponds to a particular observable trait, the number of subjects, n, may be of order 103, while the number of genetic features, p, in consideration can be of order 105.
    [Show full text]
  • Confidence Distributions and Empirical Bayes Posterior Distributions Unified As Distributions of Evidential Support David R
    Confidence distributions and empirical Bayes posterior distributions unified as distributions of evidential support David R. Bickel To cite this version: David R. Bickel. Confidence distributions and empirical Bayes posterior distributions unified asdis- tributions of evidential support. 2018. hal-01967525 HAL Id: hal-01967525 https://hal.archives-ouvertes.fr/hal-01967525 Preprint submitted on 31 Dec 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Confidence distributions and empirical Bayes posterior distributions unified as distributions of evidential support David R. Bickel December 31, 2018 Ottawa Institute of Systems Biology Department of Biochemistry, Microbiology, and Immunology Department of Mathematics and Statistics University of Ottawa 451 Smyth Road Ottawa, Ontario, K1H 8M5 +01 (613) 562-5800, ext. 8670 [email protected] Abstract While empirical Bayes methods thrive in the presence of the hundreds or thousands of simultaneous hypothesis tests in genomics and other large-scale applications, significance tests and confidence intervals
    [Show full text]
  • Objective Priors in the Empirical Bayes Framework Arxiv:1612.00064V5 [Stat.ME] 11 May 2020
    Objective Priors in the Empirical Bayes Framework Ilja Klebanov1 Alexander Sikorski1 Christof Sch¨utte1,2 Susanna R¨oblitz3 May 13, 2020 Abstract: When dealing with Bayesian inference the choice of the prior often re- mains a debatable question. Empirical Bayes methods offer a data-driven solution to this problem by estimating the prior itself from an ensemble of data. In the non- parametric case, the maximum likelihood estimate is known to overfit the data, an issue that is commonly tackled by regularization. However, the majority of regu- larizations are ad hoc choices which lack invariance under reparametrization of the model and result in inconsistent estimates for equivalent models. We introduce a non-parametric, transformation invariant estimator for the prior distribution. Being defined in terms of the missing information similar to the reference prior, it can be seen as an extension of the latter to the data-driven setting. This implies a natural interpretation as a trade-off between choosing the least informative prior and incor- porating the information provided by the data, a symbiosis between the objective and empirical Bayes methodologies. Keywords: Parameter estimation, Bayesian inference, Bayesian hierarchical model- ing, invariance, hyperprior, MPLE, reference prior, Jeffreys prior, missing informa- tion, expected information 2010 Mathematics Subject Classification: 62C12, 62G08 1 Introduction Inferring a parameter θ 2 Θ from a measurement x 2 X using Bayes' rule requires prior knowledge about θ, which is not given in many applications. This has led to a lot of controversy arXiv:1612.00064v5 [stat.ME] 11 May 2020 in the statistical community and to harsh criticism concerning the objectivity of the Bayesian approach.
    [Show full text]
  • Empirical Bayes Regression Analysis with Many Regressors but Fewer Observations
    Empirical Bayes Regression Analysis with Many Regressors but Fewer Observations Muni S. Srivastava∗and Tatsuya Kubokawa† University of Toronto and University of Tokyo December 20, 2005 In this paper, we consider the prediction problem in multiple linear regression model in which the number of predictor variables, p, is extremely large compared to the number of available observations, n. The least squares predictor based on a generalized inverse is not efficient. We propose six empirical Bayes estimators of the regression parameters. Three of them are shown to have uniformly lower prediction error than the least squares predictors when the vector of regressor variables are assumed to be random with mean t t vector zero and the covariance matrix (1/n)X X where X =(x1,... ,xn) is the p × n matrix of observations on the regressor vector centered from their sample means. For other estimators, we use simulation to show its superiority over the least squares predictor. Key words and phrases: Empirical Bayes method, James-Stein estimator, linear re- gression model, many regressors, prediction, ridge regression, shrinkage, stability. 1 Introduction In this paper, we consider the prediction problem in multiple linear regression model in which the number of predictor variables is extremely large compared to the number of observations available. More specifically, we consider the problem in which the n response variables y1,... ,yn are linearly related to the p regressor variables as y = β01n + Xβ + (1.1) t where y =(y1,... ,yn) , X is the n × p observation matrix on the p predictor variables t x1,... ,xp, centered from their sample means, 1n =(1,..
    [Show full text]
  • Application of GIS for Urban Traffic Accidents: a Critical Review
    Journal of Geographic Information System, 2019, 11, 82-96 http://www.scirp.org/journal/jgis ISSN Online: 2151-1969 ISSN Print: 2151-1950 Application of GIS for Urban Traffic Accidents: A Critical Review Niloofar Haji Mirza Aghasi Graduate School of Life and Environmental Science, Division of Spatial Information Science, University of Tsukuba, Tsukuba, Japan How to cite this paper: Aghasi, N.H.M. Abstract (2019) Application of GIS for Urban Traffic Accidents: A Critical Review. Journal of The analysis of huge data is a complex task that cannot be executed without a Geographic Information System, 11, 82-96. proper system. Geographic information systems (GISs) have been used by https://doi.org/10.4236/jgis.2019.111007 many transportation agencies and police departments to analyze and manage Received: October 15, 2018 urban traffic accident (UTA) data and for decision making aimed at decreas- Accepted: February 25, 2019 ing accident rates and increasing safety. The exact location of accidents and Published: February 28, 2019 environmental characteristics must be analyzed as UTAs occur in specific lo- Copyright © 2019 by author(s) and cations with specific characteristics. ArcGIS software is the best choice for Scientific Research Publishing Inc. obtaining meaningful information and analysis results from UTAs in an ob- This work is licensed under the Creative servational time span. GIS technology is a fundamental element for investi- Commons Attribution International gating and evaluating the complex spatial relationship among different com- License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ ponents and urban traffic accident is one of them. Micro or macro analysis of Open Access UTAs through the spatial prospective within the geographical environment and urban structure can make a deep micro understanding of UTAs patterns in addition to assisting in decision making.
    [Show full text]
  • Bayesian Variable Selection Using an Adaptive Powered Correlation Prior
    Bayesian Variable Selection Using an Adaptive Powered Correlation Prior Arun Krishna, Howard D. Bondell and Sujit K. Ghosh Department of Statistics, North Carolina State University Raleigh, NC 27695-8203, U.S.A. Correspondence Author: Howard D. Bondell email: [email protected] Telephone: (919)515-1914; Fax: (919)515-1169 Abstract The problem of selecting the correct subset of predictors within a linear model has received much attention in recent literature. Within the Bayesian framework, a popular choice of prior has been Zellner’s g-prior which is based on the inverse of empirical covariance matrix of the predictors. An extension of the Zellner’s prior is proposed in this article which allow for a power parameter on the empirical covariance of the predictors. The power parameter helps control the degree to which correlated predictors are smoothed towards or away from one another. In addition, the empirical covariance of the predictors is used to obtain suitable priors over model space. In this manner, the power parameter also helps to determine whether models containing highly collinear predictors are preferred or avoided. The proposed power parameter can be chosen via an empirical Bayes method which leads to a data adaptive choice of prior. Simulation studies and a real data example are presented to show how the power parameter is well determined from the degree of cross-correlation within predictors. The proposed modification compares favorably to the standard use of Zellner’s prior and an intrinsic prior in these examples. Keywords: Bayesian variable selection; Collinearity; Powered Correlation Prior; Zellner’s g-prior 1 1 Introduction Consider the linear regression model with n independent observations and let y = (y , ,y )0 1 · · · n be the vector of response variables.
    [Show full text]