
Bayesian Variable Selection for Predictively Optimal Regression 255
An extension of this pr ior used to address the consistency of the es timator is
referred to by [19] as the hyper-g/n prio r and defined as
π(g) =
a − 2
2n
1 +
g
n
−a/2
,
which yields the marginal likelihood
m
γ
(Y ) ∝
g
1 + g
p
γ
/2
1
1 + g
Y
⊤
P
γ
Y +
g
1 + g
Y
⊤
(I
n
− J
n
)Y
−
n−1
2
for which a close d-form expression is found to be
m
γ
(Y ) =
Γ((n − 1)/2)
√
π
(n−1)
√
n
Y
⊤
(I
n
− J
n
)Y
−
n−1
2
(1 + g)
(n−p
γ
−1)/2
[1 + g(1 − R
2
γ
)]
(n−1)/2
.
The corresponding Bayes Factor is given by
BF
γ:0
= (1 + g)
(n−p
γ
−1)/2
1 + g(1 − R
2
γ
)
−(n−1)/2
where the multiple coefficient of determination R
2
γ
for M
γ
is g iven by
1 − R
2
γ
=
Y
⊤
(I
n
− H
γ
)Y
Y
⊤
(I
n
− J
n
)Y