g-prior

Consider a data set ${\displaystyle (x_{1},y_{1}),\ldots ,(x_{n},y_{n})}$, where the ${\displaystyle x_{i}}$ are Euclidean vectors and the ${\displaystyle y_{i}}$ are scalars. The multiple regression model is formulated as

${\displaystyle y_{i}=x_{i}^{\top }\beta +\varepsilon _{i}.}$

where the ${\displaystyle \varepsilon _{i}}$ are random errors. Zellner's g-prior for ${\displaystyle \beta }$ is a multivariate normal distribution with covariance matrix proportional to the inverse Fisher information matrix for ${\displaystyle \beta }$.

Assume the ${\displaystyle \varepsilon _{i}}$ are iid normal with zero mean and variance ${\displaystyle \psi ^{-1}}$. Let ${\displaystyle X}$ be the matrix with ${\displaystyle i}$th row equal to ${\displaystyle x_{i}^{\top }}$. Then the g-prior for ${\displaystyle \beta }$ is the multivariate normal distribution with prior mean a hyperparameter ${\displaystyle \beta _{0}}$ 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.

The posterior distribution of ${\displaystyle \beta }$ 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 ${\displaystyle \beta }$. The vector of regression coefficients ${\displaystyle \beta }$ can be estimated by its posterior mean under the g-prior, i.e., as the weighted average of the maximum likelihood estimator and ${\displaystyle \beta _{0}}$,

${\displaystyle {\tilde {\beta }}=q{\hat {\beta }}+(1-q)\beta _{0}.}$

Clearly, as g →∞, the posterior mean converges to the maximum likelihood estimator.

Estimation of g is slightly less straightforward than estimation of ${\displaystyle \beta }$. A variety of methods have been proposed, including Bayes and empirical Bayes estimators.[3]