Biomedical Engineering Reference
In-Depth Information
where
p
(
x
)
=
p
(
x
|
ʱ
)
p
(
ʱ
)
d
ʱ
.
(2.71)
The estimate
x
is, then, obtained by
x
=
argmax
x
p
(
y
|
x
)
p
(
x
).
.
However, we usually have no such information and may use noninformed prior
p
To compute
p
(
x
)
using Eq. (
2.71
), we need to specify the hyperprior
p
(
ʱ
)
(
ʱ
)
=
const. Substituting this flat prior into Eq. (
2.71
), we have
p
(
x
)
∝
p
(
x
|
ʱ
)
d
ʱ
.
However, the integral in the above equation is difficult to compute. The formal
procedure to compute
p
(
x
)
in this case is to first assume the Gamma distribution for
the hyperprior
p
(
ʱ
)
, such that
N
N
)
−
1
b
a
a
−
1
e
−
b
ʱ
j
p
(
ʱ
)
=
p
(ʱ
j
)
=
1
ʓ(
a
(ʱ
j
)
.
(2.72)
j
=
1
j
=
(
)
Then,
p
x
in Eq. (
2.71
) is known to be obtained as Student
t
-distribution, such
that [
8
]
(
x
j
)
=
(
x
j
|
ʱ
j
)
(ʱ
j
)
ʱ
j
p
p
p
d
ʱ
j
2
1
/
2
exp
2
x
j
b
a
ʓ(
ʱ
j
a
−
1
e
−
b
ʱ
j
d
−
ʱ
j
=
ʱ
j
ˀ
a
)
b
−
(
a
+
2
)
x
j
2
1
b
a
ʓ(
a
+
2
)
=
√
2
+
.
(2.73)
ˀʓ(
a
)
We then assume that
a
ₒ
0 and
b
ₒ
0, (which is equivalent to making
p
(
ʱ
)
a
noninformed prior,)
p
(
x
j
)
then becomes
N
1
1
(
x
j
)
ₒ
(
)
ₒ
x
j
|
.
p
i.e.
p
x
(2.74)
|
x
j
|
|
j
=
1
Using Eq. (
2.57
), the cost function, in this case, is derived as
N
2
F(
x
)
=
ʲ
y
−
Fx
+
log
|
x
j
|
.
(2.75)
j
=
1
