5. Problems with Bayes Factors using Noninformative Priors. Consider comparing two models using Bayes factors based on noninformative priors in the setup of Section 3.6.2. The Bayes factor can be obtained using (3.34) for the two models:

(a) Consider a noninformative prior created by setting ¹j D 0, V 1 j D cIkj and letting c ! 0 for j D 1; 2. Show that the Bayes factor comparing M1 to M2 reduces to:

8>>

>>:

0 if k1 > k2 h

jX01X1j jX02 X2j i1 21¹1s2 1

¹2s2 2

N 2

if k2 D k1 1 if k1

(b) Consider a noninformative prior created by setting ¹j D 0, V1 j D 1

c 1

kj

Ikj and letting c ! 0 for j D 1; 2. Show that the Bayes factor reduces to:

(c) Consider a noninformative prior created by setting ¹j D 0, V31 j D c 1 kj
X0j Xj and letting c ! 0 for j D 1; 2. Show that the Bayes factor reduces to:
¹1s2 1 ¹2s2 2 !3N 2

X2X2 V25