Biomedical Engineering Reference
In-Depth Information
In the equations above,
R
su
and
R
uu
are obtained in the following manner. Defining
ʓ
ss
ʓ
su
ʓ
us
ʓ
uu
ʓ
−
1
=
,
(5.154)
we have
K
R
ss
R
su
R
us
R
uu
s
k
u
k
s
k
,
u
k
]
=
E
z
[
k
=
1
k
=
1
¯
K
ʓ
ss
ʓ
su
ʓ
us
ʓ
uu
s
k
k
=
1
¯
u
k
s
k
¯
s
k
¯
=
+
.
(5.155)
k
=
1
¯
s
k
k
=
1
¯
u
k
u
k
¯
u
k
¯
Therefore, we obtain:
K
K
ʓ
ss
,
s
k
R
ss
=
1
¯
s
k
¯
+
(5.156)
k
=
K
u
k
K
ʓ
su
,
R
su
=
1
¯
s
k
¯
+
(5.157)
k
=
K
u
k
K
ʓ
uu
.
R
uu
=
1
¯
u
k
¯
+
(5.158)
k
=
Also,
R
yu
is obtained as
K
u
k
.
R
yu
=
y
k
¯
(5.159)
k
=
1
5.5.3.3 Update Equations for Hyperparameters
The update equations for
ʦ
,
ʛ
, and
ʱ
are derived by maximizing the free energy,
which is rewritten as
F(
ʛ
,
ʱ
,
ʦ
)
=
E
)
[
log
p
(
y
|
z
,
A
s
)
+
log
p
(
z
)
+
log
p
(
A
)
]
,
(5.160)
(
z
,
A
where the terms
−
log
p
(
z
|
y
)
−
log
p
(
A
|
y
)
are ignored because these terms contain
none of these hyperparameters. To compute
F(
ʛ
,
ʱ
,
ʦ
)
, we use the relationships,
