Biomedical Engineering Reference
In-Depth Information
Setting the right-hand side to zero gives
K
T
1
K
ʛ
−
1
=
E
u
1
(
y
k
−
Au
k
)(
y
k
−
Au
k
)
.
(5.20)
k
=
The right-hand side is further changed to
K
T
E
u
1
(
y
k
−
Au
k
)(
y
k
−
Au
k
)
k
=
K
K
K
K
y
k
y
k
]−
y
k
u
k
]
A
T
u
k
y
k
]+
u
k
u
k
]
A
T
=
E
u
[
E
u
[
−
A
E
u
[
A
E
u
[
k
=
1
k
=
1
k
=
1
k
=
1
K
K
K
K
y
k
y
k
u
k
A
T
u
k
y
k
u
k
u
k
]
A
T
=
−
y
k
¯
−
A
¯
+
A
E
u
[
k
=
1
k
=
1
k
=
1
k
=
1
R
yu
A
T
AR
uu
A
T
=
R
yy
−
−
AR
uy
+
,
(5.21)
where
R
yy
=
k
=
1
y
k
y
k
.Using
R
yu
=
AR
uu
derived from Eq. (
5.18
), we get
T
K
R
yy
−
AR
uy
.
1
K
1
K
E
u
1
(
y
k
−
Au
k
)(
y
k
−
Au
k
)
=
(5.22)
k
=
ʛ
−
1
is a diagonal matrix, we finally derive
Considering that
1
K
ʛ
−
1
=
diag
(
R
yy
−
AR
uy
),
(5.23)
where diag
indicates a diagonal matrix obtained using the diagonal entries of a
matrix between the brackets.
(
·
)
5.2.4 Computation of Marginal Likelihood
Let us derive an expression to compute the marginal likelihood
p
(
y
|
ʸ
)
, which can be
used for monitoring the progress of the EM algorithm. Here,
ʸ
is used to collectively
express the hyperparameters
A
and
ʛ
. Using the marginalization, we obtain
p
(
y
|
ʸ
)
,
such that
∞
p
(
y
|
ʸ
)
=
p
(
y
|
u
,
ʸ
)
p
(
u
)
d
u
.
(5.24)
−∞
