Hyper nonlocal priors for variable selection in generalized linear models
type: Item
instance of: scholarly article
title: Hyper nonlocal priors for variable selection in generalized linear models
zbMATH Open document ID: 1439.62091
author: Ho-Hsiang Wu
author: Marco A. R. Ferreira
author: Mohamed Elkhouly
author: Tieming Ji
published in: Sankhyā. Series A
publication date: 2020-04-15T00:00:00Z
review text: This paper proposes a new Bayesian method of variable selection in generalized linear models. This is an important issue in any scientific discipline that uses such models in the analysis of experimental data. The following is a summary of the technical aspects of the paper. Let \(Y=(Y_,\dots,Y_{n})'\) be an \(n\)-dimensional random vector (the response variable). Let \[ X=[x_{\cdot,1},\dots,x_{\cdot,p}] \] be a numerical \(n\times p\) matrix, design matrix, where \(n\) is the number of observations and \(p\) is the total number of variables to be considered. \(j=(j_1,\dots,j_{p})\in\{0,1\}^{p}\) indicates the model \[ h(E(Y))=\beta_{0,j}+X_{j} \cdot \beta_{j} \] where \(h\) is a given link function, \(\beta_{0,j}\) is the intercept, \(X_{j}\) is the design matrix consisting only of the \(x_{\cdot,j_{k}}\), \(j_{k}=1\), \(k=1,\dots,p\) variables, and \(\beta\) is the corresponding regression vector of nonzero coefficients whose dimension is \(p_{j}\), number of ones in vector \(j\), so \(X_{j}\) is an \(n\times p_{j}\) matrix. Be \(M=\{j\vert j=(j_1,\dots,j_{p})\}\in\{0,1\}^{p}\). For each \(j\in M\), \(\beta_{0,j}\) and \(\beta_{j}\) are unknown parameters to be estimated from observations of the response variable. For each \(j\in M\), it is assumed that the random vector \(Y\) has a density given for each \(y=(y_1,\dots,y_{n})'\in\mathbb{R}^n\) by \[ f(y)=\prod_{i=1}^n\exp(((y_{i}\theta(\eta_{i,j})-b(\theta(η_{i,j})))/\varphi)+c(y_{i},\varphi)), \] where \(\theta\), \(b\), \(c\) are known functions, \[ \eta_{i,j}=h(E(Y_{i}))=\beta_{0,i,j}+X_{i,j} \cdot \beta_{j}, \] \(\eta_{j}=(\eta_{1,j},\dots,\eta_{n,j})'\) is the linear predictor vector, \(\theta(\eta_{i,j})\) is the canonical parameter for \(Y_{i}\), and \(\varphi\) is the dispersion parameter. Furthermore: \(E(Y_{i})=Db(\theta(\eta_{i,j}))\), Var\((Y_{i})=\varphi D^2b(\theta(\eta_{i,j}))\) where \(D\) is the derivative operator, and there are all the higher-order derivatives. The problem of choosing the best …
Mathematics Subject Classification ID: 62F15
Mathematics Subject Classification ID: 62F07
Mathematics Subject Classification ID: 62J12
Mathematics Subject Classification ID: 62P10
zbMATH DE Number: 7189653
zbMATH Keywords: Bayesian variable selection
zbMATH Keywords: generalized linear model
zbMATH Keywords: nonlocal prior
zbMATH Keywords: scale mixtures
zbMATH Keywords: variable selection consistency
reviewed by: Oscar Bustos
describes a project that uses: glmnet
describes a project that uses: BVSNLP
describes a project that uses: GA
describes a project that uses: R
describes a project that uses: UCI-ml
describes a project that uses: Fahrmeir
MaRDI profile type: Publication
full work available at URL: s13171-018-0151-9
OpenAlex ID: W2901199062
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Wikidata QID: Q128984486
DOI: 10.1007/S13171-018-0151-9
description: scientific article; zbMATH DE number 7189653