View source for G-prior

{{Short description|Type of probability distribution used in statistics}}
{{lowercase title}}
In [[statistics]], the '''g-prior''' is an [[objective prior]] for the regression coefficients of a [[multiple regression]]. It was introduced by [[Arnold Zellner]].<ref>{{cite book |last=Zellner |first=A. |year=1986 |chapter=On Assessing Prior Distributions and Bayesian Regression Analysis with g Prior Distributions |title=Bayesian Inference and Decision Techniques: Essays in Honor of Bruno de Finetti |editor-first=P. |editor-last=Goel |editor2-first=A. |editor2-last=Zellner |series=Studies in Bayesian Econometrics and Statistics |volume=6 |location=New York |publisher=Elsevier |pages=233–243 |isbn=978-0-444-87712-3 }}</ref>
It is a key tool in [[Bayes estimator|Bayes]] and [[empirical Bayes]] [[variable selection]].<ref>{{cite journal |last=George |first=E. |last2=Foster |first2=D. P. |year=2000 |title=Calibration and empirical Bayes variable selection |journal=[[Biometrika]] |volume=87 |issue=4 |pages=731–747 |doi=10.1093/biomet/87.4.731 |citeseerx=10.1.1.18.3731 }}</ref><ref name="gpriormix">{{cite journal |last=Liang |first=F. |last2=Paulo |first2=R. |last3=Molina |first3=G. |last4=Clyde |first4=M. A. |last5=Berger |first5=J. O. |year=2008 |title=Mixtures of g priors for Bayesian variable selection |journal=[[Journal of the American Statistical Association]] |volume=103 |issue=481 |pages=410–423 |doi=10.1198/016214507000001337 |citeseerx=10.1.1.206.235 }}</ref>

==Definition==

Consider a data set <math>(x_1,y_1),\ldots,(x_n,y_n)</math>, where the <math>x_i</math> are [[Euclidean vector]]s and the <math>y_i</math> are [[Scalar (mathematics)|scalar]]s.
The multiple regression model is formulated as
:<math>y_i = x_i^\top\beta + \varepsilon_i.</math>
where the <math>\varepsilon_i</math> are random errors.
Zellner's g-prior for <math>\beta</math> is a [[multivariate normal distribution]] with covariance matrix proportional to the inverse [[Fisher information]] matrix for <math>\beta</math>, similar to a [[Jeffreys prior]].

Assume the <math>\varepsilon_i</math> are [[Independent and identically distributed random variables|i.i.d.]] normal with zero mean and variance <math>\psi^{-1}</math>. Let <math>X</math> be the matrix with <math>i</math>th row equal to <math>x_i^\top</math>.
Then the g-prior for <math>\beta</math> is the multivariate normal distribution with prior mean a hyperparameter <math>\beta_0</math> and covariance matrix proportional to <math>\psi^{-1}(X^\top X)^{-1}</math>, i.e.,
:<math>\beta |\psi  \sim \text{N}[\beta_0,g\psi^{-1} (X^\top X)^{-1}].</math>
where g is a positive scalar parameter.

==Posterior distribution of beta ==

The posterior distribution of <math>\beta</math> is given as
:<math>\beta |\psi,x,y  \sim \text{N}\Big[q\hat\beta+(1-q)\beta_0,\frac q\psi(X^\top X)^{-1}\Big].</math>
where <math>q=g/(1+g)</math> and
:<math>\hat\beta = (X^\top X)^{-1}X^\top y.</math>
is the maximum likelihood (least squares) estimator of <math>\beta</math>. The vector of regression coefficients <math>\beta</math> can be estimated by its posterior mean under the g-prior, i.e., as the weighted average of the maximum likelihood estimator and <math>\beta_0</math>,
:<math>\tilde\beta = q\hat\beta+(1-q)\beta_0.</math>
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 <math>\beta</math>.
A variety of methods have been proposed, including Bayes and empirical Bayes estimators.<ref <ref name="gpriormix"></ref>

==References==
{{reflist}}

==Further reading==
* {{cite book |first=Jyotishka |last=Datta |first2=Jayanta K. |last2=Ghosh |year=2015 |chapter=In Search of Optimal Objective Priors for Model Selection and Estimation |pages=225–243 |title=Current Trends in Bayesian Methodology with Applications |editor1-first=Satyanshu Kumar |editor1-last=Upadhyay |editor2-first=Umesh |editor2-last=Singh |editor3-first=Dipak K. |editor3-last=Dey |editor4-first=Appaia |editor4-last=Loganathan |display-editors=1 |location= |publisher=CRC Press |isbn=978-1-4822-3511-1 }}
* {{cite book |first=Jean-Michel |last=Marin |first2=Christian P. |last2=Robert |chapter=Regression and Variable Selection |pages=47–84 |title=Bayesian Core : A Practical Approach to Computational Bayesian Statistics |location=New York |publisher=Springer |year=2007 |isbn=978-0-387-38979-0 |doi=10.1007/978-0-387-38983-7_3 }}

[[Category:Bayesian statistics]]
[[Category:Regression analysis]]