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.^[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 $\beta$ , similar to a Jeffreys prior.

Assume the $\varepsilon _{i}$ are i.i.d. normal with zero mean and variance $\psi ^{-1}$ . Let $X$ be the matrix with $i$ th 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 $\psi ^{-1}(X^{\top }X)^{-1}$ , i.e.,

\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

\beta |\psi ,x,y\sim {\text{N}}{\Big [}q{\hat {\beta }}+(1-q)\beta _{0},{\frac {q}{\psi }}(X^{\top }X)^{-1}{\Big ]}.

where $q=g/(1+g)$ and

{\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 $\beta _{0}$ ,

{\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. Vol. 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.
^ ^a ^b 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.

Definition

Posterior distribution of beta

Selection of g

References

Further reading