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About 7,778 papers for “Bayesian linear regression”

Bayesian Genomic Linear Regression

Osval A. Montesinos‐López, Abelardo Montesinos‐López, José Crossa · 2022

Abstract The Bayesian paradigm for parameter estimation is introduced and linked to the main problem of genomic-enabled prediction to predict the trait of interest of the non-phenotyped individuals from genotypic information, environment variables, or other information (covariates). In this situation, a convenient practice is to include the individuals to be predicted in the posterior distribution to be sampled. We explained how the Bayesian Ridge regression method is derived and exemplified with data from plant breeding genomic selection. Other Bayesian methods (Bayes A, Bayes B, Bayes C, and Bayesian Lasso) were also described and exemplified for genome-based prediction. The chapter presented several examples that were implemented in the Bayesian generalized linear regression (BGLR) library for continuous response variables. The predictor under all these Bayesian methods includes main effects (of environments and genotypes) as well as interaction terms related to genotype × environment interaction.

Cited by 2Open access

Functional Bayesian linear regression

Paul-Marie Grollemund · theses.fr (ABES) · 2017

Un outil fondamental en statistique est le modèle de régression linéaire. Lorsqu'une des covariables est une fonction, on fait face à un problème de statistique en grande dimension. Pour conduire l'inférence dans cette situation, le modèle doit être parcimonieux, par exemple en projetant la covariable fonctionnelle dans des espaces de plus petites dimensions. Dans cette thèse, nous proposons une approche bayésienne nommée Bliss pour ajuster le modèle de régression linéaire fonctionnel. Notre modèle, plus précisément la distribution a priori, suppose que la fonction coefficient est une fonction en escalier. A partir de la distribution a posteriori, nous définissons plusieurs estimateurs bayésiens, à choisir suivant le contexte : un estimateur du support et deux estimateurs, un lisse et un estimateur constant par morceaux. A titre d'exemple, nous considérons un problème de prédiction de la production de truffes noires du Périgord en fonction d'une covariable fonctionnelle représentant l'évolution des précipitations au cours du temps. En terme d'impact sur les productions, la méthode Bliss dégage alors deux périodes de temps importantes pour le développement de la truffe. Un autre atout du paradigme bayésien est de pouvoir inclure de l'information dans la loi a priori, par exemple l'expertise des trufficulteurs et des biologistes sur le développement de la truffe. Dans ce but, nous proposons deux variantes de la méthode Bliss pour prendre en compte ces avis. La première variante récolte de manière indirecte l'avis des experts en leur proposant de construire des données fictives. La loi a priori correspond alors à la distribution a posteriori sachant ces pseudo-données. En outre, un système de poids relativise l'impact de chaque expert ainsi que leurs corrélations. La seconde variante récolte explicitement l'avis des experts sur les périodes de temps les plus influentes sur la production et si cet l'impact est positif ou négatif. La construction de la loi a priori repose alors sur une pénalisation des fonctions coefficients en contradiction avec ces avis. Enfin, ces travaux de thèse s'attachent à l'analyse et la compréhension du comportement de la méthode Bliss. La validité de l'approche est justifiée par une étude asymptotique de la distribution a posteriori. Nous avons construit un jeu d'hypothèses spécifique au modèle Bliss, pour écrire une démonstration efficace d'un théorème de Wald. Une des difficultés est la mauvaise spécification du modèle Bliss, dans le sens où la vraie fonction coefficient n'est sûrement pas une fonction en escalier. Nous montrons que la loi a posteriori se concentre autour d'une fonction coefficient en escalier, obtenue par projection au sens de la divergence de Kullback-Leibler de la vraie fonction coefficient sur un ensemble de fonctions en escalier. Nous caractérisons cette fonction en escalier à partir du design et de la vraie fonction coefficient.

Cited by 0

Bayesian Linear Regression Model

Biliana S. Güner, Svetlozar T. Rachev, J.S. Hsu et al. · Encyclopedia of Financial Models · 2012

Linear regression is the “workhorse” of financial modeling. Cornerstone applications, such as asset pricing models, as well as time series models, are built around linear regression's methods and tools. Casting the linear regression methodology in a Bayesian setting helps account for estimation uncertainty, allows for integration of prior information, and makes accessible the Bayesian numerical simulation framework. Keywords: Bayesian linear regression; diffuse; prior; informative; prior; Gibbs sampler

Cited by 1

Safe-Bayesian generalized linear regression

Heide, R. de, Kirichenko, A., Mehta, N. et al. · Centrum Wiskunde & Informatica (CWI), the national research institute for mathematics and computer science in the Netherlands · 2020

We study generalized Bayesian inference under misspecification, i.e. when the model is ‘wrong but useful’. Generalized Bayes equips the likelihood with a learning rate η. We show that for generalized linear models (GLMs), η-generalized Bayes concentrates around the best approximation of the truth within the model for specific ηeq1, even under severely misspecified noise, as long as the tails of the true distribution are exponential. We derive MCMC samplers for generalized Bayesian lasso and logistic regression and give examples of both simulated and real-world data in which generalized Bayes substantially outperforms standard Bayes.

Cited by 0Open access

Bayesian Linear Regression with Standardized

2012

This example uses the MCMC procedure to fit a Bayesian linear regression model with standardized covariates. It shows how the random walk Metropolis sampling algorithm struggles when the scales of the regression parameters are vastly different. It also illustrates that the sampling algorithm performs quite well when the covariates are standardized. The SAS source code for this example is available as an attachment in the text file. In Adobe Acrobat, right-click the icon in the margin and select Save Embedded File to Disk. You can also double-click to open the file immediately. Analysis The following data set contains salary and performance information for Major League Baseball players (excluding pitchers) who played at least one game in both the 1986 and 1987 seasons. The

Cited by 0

Safe-Bayesian Generalized Linear Regression

Rianne de Heide, Alisa Kirichenko, Nishant A. Mehta et al. · arXiv (Cornell University) · 2019

We study generalized Bayesian inference under misspecification, i.e. when the model is 'wrong but useful'. Generalized Bayes equips the likelihood with a learning rate $η$. We show that for generalized linear models (GLMs), $η$-generalized Bayes concentrates around the best approximation of the truth within the model for specific $η\neq 1$, even under severely misspecified noise, as long as the tails of the true distribution are exponential. We derive MCMC samplers for generalized Bayesian lasso and logistic regression and give examples of both simulated and real-world data in which generalized Bayes substantially outperforms standard Bayes.

Cited by 18Open access

Interpretable Bayesian Functional Linear Regression

Paul-Marie Grollemund, Christophe Abraham, Meïli Baragatti et al. · HAL (Le Centre pour la Communication Scientifique Directe) · 2015

Interpretable Bayesian Functional Linear Regression. 47. Journees de Statistique de la SFdS

Cited by 0

Bayesian Piecewise Linear Regression Using Multivariate Linear Splines

Holmes And, C C Holmes, B K Mallick · 2007

We present a Bayesian analysis of a piecewise linear model constructed using a basis function approach with a multivariate generalization of the univariate linear spline. Prior distributions are adopted on both the number and locations of the splines. Conditioning on the data produces a Bayes local linear model with distributions on predictions and on local linear parameters that vary smoothly over the design space. The method is spatially adaptive and fairly automatic and the difficult problem of "bandwidth" selection does not arise. Furthermore, by using splines of lower dimension than the data the model automatically removes irrelevant covariates and determines the relative importance of those remaining. KEYWORDS: Bayesian linear regression; nonlinear regression; multivariate splines; piecewise linear; local linear regression. 1 Introduction Many methods exist for modelling the mean regression surface, E(Y jX), of a response variable Y , given a design matrix of covariates or pre...

Cited by 0

Bayesian Variable Selection in Linear Regression

Toby J. Mitchell, John J. Beauchamp · Journal of the American Statistical Association · 1988

Abstract This article is concerned with the selection of subsets of predictor variables in a linear regression model for the prediction of a dependent variable. It is based on a Bayesian approach, intended to be as objective as possible. A probability distribution is first assigned to the dependent variable through the specification of a family of prior distributions for the unknown parameters in the regression model. The method is not fully Bayesian, however, because the ultimate choice of prior distribution from this family is affected by the data. It is assumed that the predictors represent distinct observables; the corresponding regression coefficients are assigned independent prior distributions. For each regression coefficient subject to deletion from the model, the prior distribution is a mixture of a point mass at 0 and a diffuse uniform distribution elsewhere, that is, a “spike and slab” distribution. The random error component is assigned a normal distribution with mean 0 and standard deviation σ, where ln(σ) has a locally uniform noninformative prior distribution. The appropriate posterior probabilities are derived for each submodel. If the regression coefficients have identical priors, the posterior distribution depends only on the data and the parameter γ, which is the height of the spike divided by the height of the slab for the common prior distribution. This parameter is not assigned a probability distribution; instead, it is considered a parameter that indexes the members of a class of Bayesian methods. Graphical methods are proposed as informal guides for choosing γ, assessing the complexity of the response function and the strength of the individual predictor variables, and assessing the degree of uncertainty about the best submodel. The following plots against γ are suggested: (a) posterior probability that a particular regression coefficient is 0; (b) posterior expected number of terms in the model; (c) posterior entropy of the submodel distribution; (d) posterior predictive error; and (e) posterior probability of goodness of fit. Plots (d) and (e) are suggested as ways to choose y. The predictive error is determined using a Bayesian cross-validation approach that generates a predictive density for each observation, given all of the data except that observation, that is, a type of “leave one out” approach. The goodness-of-fit measure is the sum of the posterior probabilities of all submodels that pass a standard F test for goodness of fit relative to the full model, at a specified level of significance. The dependence of the results on the scaling of the variables is discussed, and some ways to choose the scaling constants are suggested. Examples based on a large data set arising from an energy-conservation study are given to demonstrate the application of the methods.

Cited by 180

Linear regression in the Bayesian framework

Thierry A. Mara · ArXiv.org · 2019

These notes aim at clarifying different strategies to perform linear regression from given dataset. Methods like the weighted and ordinary least squares, ridge regression or LASSO are proposed in the literature. The present article is my understanding of these methods which are, according to me, better unified in the Bayesian framework. The formulas to address linear regression with these methods are derived. The KIC for model selection is also derived in the end of the document.

Cited by 1Open access

A Note on Bayesian Linear Regression

Mohammad Shekaramiz, Todd K. Moon · Digital Commons - USU (Utah State University) · 2019

In this report, we briefly discuss Bayesian linear regression as well as the proof for the inference to perform prediction based on the training data using this technique.

Cited by 0Open access

Bayesian Variable Selection in Linear Regression

Toby J. Mitchell, John J. Beauchamp · Journal of the American Statistical Association · 1988

Abstract This article is concerned with the selection of subsets of predictor variables in a linear regression model for the prediction of a dependent variable. It is based on a Bayesian approach, intended to be as objective as possible. A probability distribution is first assigned to the dependent variable through the specification of a family of prior distributions for the unknown parameters in the regression model. The method is not fully Bayesian, however, because the ultimate choice of prior distribution from this family is affected by the data. It is assumed that the predictors represent distinct observables; the corresponding regression coefficients are assigned independent prior distributions. For each regression coefficient subject to deletion from the model, the prior distribution is a mixture of a point mass at 0 and a diffuse uniform distribution elsewhere, that is, a “spike and slab” distribution. The random error component is assigned a normal distribution with mean 0 and standard deviation σ, where ln(σ) has a locally uniform noninformative prior distribution. The appropriate posterior probabilities are derived for each submodel. If the regression coefficients have identical priors, the posterior distribution depends only on the data and the parameter γ, which is the height of the spike divided by the height of the slab for the common prior distribution. This parameter is not assigned a probability distribution; instead, it is considered a parameter that indexes the members of a class of Bayesian methods. Graphical methods are proposed as informal guides for choosing γ, assessing the complexity of the response function and the strength of the individual predictor variables, and assessing the degree of uncertainty about the best submodel. The following plots against γ are suggested: (a) posterior probability that a particular regression coefficient is 0; (b) posterior expected number of terms in the model; (c) posterior entropy of the submodel distribution; (d) posterior predictive error; and (e) posterior probability of goodness of fit. Plots (d) and (e) are suggested as ways to choose y. The predictive error is determined using a Bayesian cross-validation approach that generates a predictive density for each observation, given all of the data except that observation, that is, a type of “leave one out” approach. The goodness-of-fit measure is the sum of the posterior probabilities of all submodels that pass a standard F test for goodness of fit relative to the full model, at a specified level of significance. The dependence of the results on the scaling of the variables is discussed, and some ways to choose the scaling constants are suggested. Examples based on a large data set arising from an energy-conservation study are given to demonstrate the application of the methods.

Cited by 1,394