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On assessing prior distributions and Bayesian regression analysis with $$g$$-prior distributions. (English) Zbl 0655.62071
Bayesian inference and decision techniques, Essays Hon. Bruno de Finetti, Stud. Bayesian Econ. Stat. 6, 233-243 (1986).
[For the entire collection see Zbl 0608.00012.]
Considered is the normal linear multiple regression model $$y=X\beta +u$$, where $$u$$ is $$N(0,\sigma 2I_ n)$$-distributed. Several approaches to use prior information on the parameters $$\beta$$ and $$\sigma$$ for estimating $$\beta$$ are discussed. As a new approach to the problem the author proposes to use Muth’s rational expectations hypothesis and an imaginary sample based on the model above with u replaced by an $$N(0,g^{-1}\sigma 2I_ n)$$-distributed error, $$0<g<\infty$$, to get a reference informative prior distribution. It turns out that this is in the inverted-gamma-normal form. For a particular version the posterior distributions are then calculated and sampling properties of the posterior mean are derived.
The approach is shown to be equivalent to a least squares approach where $$\beta$$ is restricted to lie in an ellipsoid with prescribed center and magnitude and determined by the matrix $$X'X$$.

##### MSC:
 62J05 Linear regression; mixed models 62F15 Bayesian inference 62C10 Bayesian problems; characterization of Bayes procedures