Let M denote the model Y = /?X + u and consider comparing &I to the niodel where M' : /I = 0.

13.3 Using a noninfonnative prior distribution for 02, p(a2) = l / d , and Zeliner's g-prior for /?. p - N ( 0 , o*g(X'X)-' ), find the marginal like-

Ilhood for Y under model JM.

Find the marginal likelihood for Y under the model M* : )I = 0 (use the same noninformativc prior for at).

Using equal prior probabilities on M and M*,sho w that the po2terior odds of ,U*to M goes to a nonzero constant (1 + g)' '('-') as /3 goes to infinity. (Even though one becomes certain that ,&iIs "w rong, the posterior odds or Bayes' factor does not go to zero, which has led many to reject g-priors for model selection; this criticism applies equally for model averaging; see Berger and Pericchi (2001 ) for alternatives.)

Using the marginal likelihood under M, find the maximum likelihood estimate of g. d . Using the Empirical Bayes g-prior substituted for g in the g-prior), what happens with the limit'?

For model M, consider a Cauchy prior for 1 c2 Zellner and Siow, 1980) of the form rather than the conjugate normal. Show that the Cauchy distribution can be written as a scale mixture of normal distributions, where # 1 u2, 1 is N ( 0 , n*/A(X'X)-') and i. has a gamma(0.5, 2) distribution (parameterized so that the mean is 1).

Investigate the limiting behavior of the posterior odds under the Cauchy prior distribution. Using the scale mix-htre of normals representation, the marginal can be obtained using one-dimensional numerical integration.