In statistics, the g-prior is an objective prior for the regression coefficients of a multiple regression. It was introduced by Arnold Zellner. It is a key tool in Bayes and empirical Bayes variable selection.
Definition Consider a data set ( x 1 , y 1 ) , … , ( x n , y n ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{n},y_{n})} , where the x i {\displaystyle x_{i}} are Euclidean vectors and the y i {\displaystyle y_{i}} are scalars. The multiple regression model is formulated as
y i = x i ⊤ β + ε i . {\displaystyle y_{i}=x_{i}^{\top }\beta +\varepsilon _{i}.}
where the ε i {\displaystyle \varepsilon _{i}} are random errors. Zellner's g-prior for β {\displaystyle \beta } is a multivariate normal distribution with covariance matrix proportional to the inverse Fisher information matrix for β {\displaystyle \beta } , similar to a Jeffreys prior. Assume the ε i {\displaystyle \varepsilon _{i}} are i.i.d. normal with zero mean and variance ψ − 1 {\displaystyle \psi ^{-1}} . Let X {\displaystyle X} be the matrix with i {\displaystyle i} th row equal to x i ⊤ {\displaystyle x_{i}^{\top }} . Then the g-prior for β {\displaystyle \beta } is the multivariate normal distribution with prior mean a hyperparameter β 0 {\displaystyle \beta _{0}} and covariance matrix proportional to ψ − 1 ( X ⊤ X ) − 1 {\displaystyle \psi ^{-1}(X^{\top }X)^{-1}} , i.e.,
β | ψ ∼ N [ β 0 , g ψ − 1 ( X ⊤ X ) − 1 ] . {\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 β {\displaystyle \beta } is given as
β | ψ , x , y ∼ N [ q β ^ + ( 1 − q ) β 0 , q ψ ( X ⊤ X ) − 1 ] . {\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 q = g / ( 1 + g ) {\displaystyle q=g/(1+g)} and
β ^ = ( X ⊤ X ) − 1 X ⊤ y . {\displaystyle {\hat {\beta }}=(X^{\top }X)^{-1}X^{\top }y.}
is the maximum likelihood (least squares) estimator of β {\displaystyle \beta } . The vector of regression coefficients β {\displaystyle \beta } can be estimated by its posterior mean under the g-prior, i.e., as the weighted average of the maximum likelihood estimator and β 0 {\displaystyle \beta _{0}} ,
β ~ = q β ^ + ( 1 − q ) β 0 . {\displaystyle {\tilde {\beta }}=q{\hat {\beta }}+(1-q)\beta _{0}.}
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