Biomedical Engineering Reference
In-Depth Information
L
2
ʛ
−
1
1
2
diag
A
T
=
−
[
A
ʱ
]
.
(5.81)
Then, using Eq. (
5.80
), we derive the derivative of
F(
ʛ
,
ʱ
)
with respect to
ʛ
such that
∂
∂
ʛ
F(
ʛ
,
ʱ
)
K
2
ʛ
−
1
A
T
K
1
2
L
2
ʛ
−
1
1
2
A
T
=
E
−
1
(
y
k
−
Au
k
)(
y
k
−
Au
k
)
+
−
ʱ
(
A
,
u
)
k
=
K
2
ʛ
−
1
1
2
E
A
[−
R
yu
A
T
AR
uu
A
T
ʛ
−
1
A
T
=
+
R
yy
+
+
AR
uy
−
+
L
−
A
ʱ
]
K
2
ʛ
−
1
1
2
(
L
2
ʛ
−
1
1
2
E
A
[
R
yu
¯
A
T
−
¯
AR
uy
)
+
A
T
=
−
R
yy
−
−
AR
uu
A
+
A
ʱ
]
.
(5.82)
The following relationship holds:
A
T
A
T
A
T
E
A
[
AR
uu
A
+
A
ʱ
]=
E
A
[
A
(
R
uu
+
ʱ
)
]=
E
A
[
A
ʨ
]
(5.83)
Also, we can prove the relationship
ʨ
¯
A
T
R
yu
¯
A
T
A
T
]=
¯
A
ʛ
−
1
ʛ
−
1
E
A
[
A
ʨ
+
L
=
+
L
.
(5.84)
The proof of this equation is provided in Sect.
5.7.1
. Thus, the relationship
∂
∂
ʛ
F(
ʛ
,
ʱ
)
=
K
2
ʛ
−
1
1
2
(
R
yu
¯
A
T
−
¯
AR
uy
)
−
R
yy
−
L
2
ʛ
−
1
1
2
(
K
2
ʛ
−
1
1
2
(
R
yu
¯
A
T
ʛ
−
1
R
yy
−
¯
AR
uy
)
(5.85)
+
−
+
L
)
=
−
holds. Setting the right-hand side of the equation above zero, we obtain the update
equation for
ʛ
as
1
K
ʛ
−
1
R
yy
−
¯
AR
uy
).
=
diag
(
(5.86)
5.3.3 Computation of Free Energy
Although we cannot compute the exact marginal likelihood in the VBEM algorithm,
we can compute its lower bound using the free energy. The free energy after the
E-step of the VBEM algorithm is expressed as
