Proof. Due to separation,
p(y| )
is monotonic increasing in s to a limit
L
, so
that
lim
s
!1
p(y|
s
) = L
. By Bayes’ rule,
p( |y) =
p(y| )p( | )
1
R
1
p(y| )p( | )d
=
p(y| )p( | )
p(y| )
| {z }
constant w.r.t.
.
Integrating out the other parameters s
= h
cons
, 1, 2, ...,
k
i
to obtain the
posterior distribution of s,
p(
s
|y) =
1
R
1
p(y| )p( | )d
s
p(y| )
,
(1)
and the prior distribution of s,
p(
s
| ) =
1
Z
1
p( | )d
s
.
Notice that
p(
s
|y) / p(
s
| )
iff
p(
s
|y)
p( | )
= k
, where the constant
k 6= 0
.Thus,