g-prior

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. 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 where the ε i {displaystyle varepsilon _{i}} are random errors.Zellner's g-prior for β {displaystyle eta } is a multivariate normal distribution with covariance matrix proportional to the inverse Fisher information matrix for β {displaystyle eta } . Assume the ε i {displaystyle varepsilon _{i}} are iid 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}^{ op }} .Then the g-prior for β {displaystyle eta } is the multivariate normal distribution with prior mean a hyperparameter β 0 {displaystyle eta _{0}} and covariance matrix proportional to ψ − 1 ( X ⊤ X ) − 1 {displaystyle psi ^{-1}(X^{ op }X)^{-1}} , i.e.,