In statistics, the g-prior is an objective prior for the regression coefficients of a multiple regression. It was introduced by Arnold Zellner.[1] It is a key tool in Bayes and empirical Bayes variable selection.[2][3]
Definition
Consider a data set \( (x_{1},y_{1}),\ldots ,(x_{n},y_{n}) \), where the \( x_{i} \) are Euclidean vectors and the \( y_{i} \) are scalars. The multiple regression model is formulated as
\( y_{i}=x_{i}^{\top }\beta +\varepsilon _{i}. \)
where the \( \varepsilon _{i} \) are random errors. Zellner's g-prior for \( \beta \) is a multivariate normal distribution with covariance matrix proportional to the inverse Fisher information matrix for\( \beta \) .
Assume the \( \varepsilon _{i} \) are iid normal with zero mean and variance \( \psi^{-1} \). Let X be the matrix with ith row equal to \( x_{i}^{\top } \). Then the g-prior for \( \beta \) is the multivariate normal distribution with prior mean a hyperparameter \( \beta _{0} \) and covariance matrix proportional to \( {\displaystyle \psi ^{-1}(X^{\top }X)^{-1}} \), i.e.,
\( {\displaystyle \beta |\psi \sim {\text{N}}[\beta _{0},g\psi ^{-1}(X^{\top }X)^{-1}].} \)
where g is a positive scalar parameter.
Posterior distribution of \( \beta \)
The posterior distribution of \( \beta \) is given as
\( {\displaystyle \beta |\psi ,x,y\sim {\text{N}}{\Big [}q{\hat {\beta }}+(1-q)\beta _{0},{\frac {q}{\psi }}(X^{\top }X)^{-1}{\Big ]}.} \)
where \( {\displaystyle q=g/(1+g)} \) and
\( {\displaystyle {\hat {\beta }}=(X^{\top }X)^{-1}X^{\top }y.}
is the maximum likelihood (least squares) estimator of \( \beta \) . The vector of regression coefficients \( \beta \) can be estimated by its posterior mean under the g-prior, i.e., as the weighted average of the maximum likelihood estimator and \( \beta _{0} \) ,
\( {\displaystyle {\tilde {\beta }}=q{\hat {\beta }}+(1-q)\beta _{0}.} \)
Clearly, as g →∞, the posterior mean converges to the maximum likelihood estimator.
Selection of g
Estimation of g is slightly less straightforward than estimation of \( \beta \) . A variety of methods have been proposed, including Bayes and empirical Bayes estimators.[3]
References
Zellner, A. (1986). "On Assessing Prior Distributions and Bayesian Regression Analysis with g Prior Distributions". In Goel, P.; Zellner, A. (eds.). Bayesian Inference and Decision Techniques: Essays in Honor of Bruno de Finetti. Studies in Bayesian Econometrics and Statistics. 6. New York: Elsevier. pp. 233–243. ISBN 978-0-444-87712-3.
George, E.; Foster, D. P. (2000). "Calibration and empirical Bayes variable selection". Biometrika. 87 (4): 731–747. CiteSeerX 10.1.1.18.3731. doi:10.1093/biomet/87.4.731.
Liang, F.; Paulo, R.; Molina, G.; Clyde, M. A.; Berger, J. O. (2008). "Mixtures of g priors for Bayesian variable selection". Journal of the American Statistical Association. 103 (481): 410–423. CiteSeerX 10.1.1.206.235. doi:10.1198/016214507000001337.
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