Biomedical Engineering Reference
In-Depth Information
¯
ʓ
where
is the posterior precisionmatrix. The exponential
part of the above Gaussian distribution is given by
x
is the posterior mean, and
1
2
(
1
2
x
k
ʓ
T
x
k
ʓ
¯
−
x
k
− ¯
x
k
)
ʓ
(
x
k
− ¯
x
k
)
=−
x
k
+
x
k
+
C,
(B.21)
where
represents terms that do not contain
x
k
. The exponential part of the right-hand
side of Eq. (
B.19
) is given by
C
x
k
ʦ
1
2
T
−
x
k
+
(
y
k
−
Hx
k
)
ʛ
(
y
k
−
Hx
k
)
.
(B.22)
The above expression can be rewritten as
1
2
x
k
(
ʦ
+
H
T
x
k
H
T
y
k
+
C
,
−
ʛ
H
)
x
k
+
ʛ
(B.23)
C
again represents terms that do not contain
x
k
. Comparing the quadratic and
linear terms of
x
k
between Eqs. (
B.21
) and (
B.23
) gives the relationships
where
H
T
ʓ
=
ʦ
+
ʛ
,
H
(B.24)
x
k
=
ʓ
−
1
H
T
H
T
)
−
1
H
T
¯
ʛ
y
k
=
(
ʦ
+
ʛ
H
ʛ
y
k
.
(B.25)
The precision matrix and the mean of the posterior distribution are obtained in
Eqs. (
B.24
) and (
B.25
), respectively. This
x
k
is the MMSE (and also MAP) estimate
of
x
k
. Using the matrix inversion formula in Eq. (
C.92
),
¯
x
k
can also be expressed as
¯
x
k
=
ʦ
−
1
H
T
ʦ
−
1
H
T
+
ʛ
−
1
)
−
1
y
k
=
ʥ
H
T
ʣ
−
1
y
¯
(
H
y
k
,
(B.26)
ʦ
−
1
, is the covariance matrix of the prior distribution,
where
ʥ
, which is equal to
and
ʣ
y
is expressed in Eq. (
B.30
).
B.4
Derivation of the Marginal Distribution
p
(
y
)
The probability for the observation
y
is obtained using
p
p
K
K
K
p
(
y
)
=
p
(
y
k
)
=
(
x
k
,
y
k
)
d
x
k
=
(
y
k
|
x
k
)
p
(
x
k
)
d
x
k
.
(B.27)
k
=
1
k
=
1
k
=
1
Substituting Eqs. (
B.16
) and (
B.14
) into the equation above, we get
