Biomedical Engineering Reference
In-Depth Information
Therefore, we have
M
∂
∂
1
2
1
ʻ
j
a
j
ʱ
a
j
=
ʛ
A
ʱ
.
(5.65)
A
j
=
We also have
K
K
∂
∂
1
2
(
T
u
k
.
1
−
y
k
−
Au
k
)
ʛ
(
y
k
−
Au
k
)
=
ʛ
1
(
y
k
−
Au
k
)
(5.66)
A
k
=
k
=
Consequently, we can derive
K
∂
∂
u
k
A
log
p
(
A
|
y
)
=
E
u
ʛ
1
(
y
k
−
Au
k
)
−
ʛ
A
ʱ
k
=
=
ʛ
R
yu
−
ʛ
A
(
R
uu
+
ʱ
).
(5.67)
Setting the right-hand side of the equation above to zero, we get
¯
A
R
uu
+
ʱ
)
−
1
=
R
yu
(
.
(5.68)
where
R
uu
and
R
yu
are defined in Eqs. (
5.15
) and (
5.16
). The precision
is
obtained as the coefficient of
a
j
in the right-hand side of Eq. (
5.67
). The second term
in the right-hand side of this equation can be rewritten as
ʻ
j
ʨ
⊡
⊣
⊤
⊦
(
⊡
⊣
⊤
⊦
.
ʻ
1
a
1
.
ʻ
M
a
T
M
ʻ
1
a
1
(
R
uu
+
ʱ
)
.
ʛ
(
R
uu
+
ʱ
)
=
R
uu
+
ʱ
)
=
A
(5.69)
ʻ
M
a
T
M
(
R
uu
+
ʱ
)
Thus,
ʨ
is obtained as
ʨ
=
R
uu
+
ʱ
.
(5.70)
Equations (
5.68
) and (
5.70
) are the M-step update equations in the VBFA algorithm.
However, to compute these equations, we need to know the hyperparameters
ʱ
and
ʛ
. The next subsection deals with the estimation of
ʱ
.
5.3.2.3 Update Equation for Hyperparameter
ʱ
is estimated by maximizing the free energy.
3
The hyperparameter
ʱ
According to
the arguments in Sect. B.6, the free energy is expressed as
3
An estimate that maximizes the free energy is the MAP estimate under the assumption of the
non-informative prior for the hyperparameter.
