Biomedical Engineering Reference
In-Depth Information
E
T
x
=
argmin
(
x
−
x
)
(
x
−
x
)
.
(B.7)
x
This estimate is called the minimum mean squared error (MMSE) estimate. Here,
taking the relationship,
∞
E
T
T
(
x
−
x
)
(
x
−
x
)
=
(
x
−
x
)
(
x
−
x
)
p
(
x
,
y
)
d
x
d
y
−∞
∞
∞
d
x
p
T
=
(
x
−
x
)
(
x
−
x
)
p
(
x
|
y
)
(
y
)
d
y
,
(B.8)
−∞
−∞
T
and the fact that
p
(
y
)
≥
0 into consideration, the
x
that minimizes
E
[
(
x
−
x
)
(
x
−
x
)
]
is equal to
∞
−∞
(
T
x
=
argmin
x
x
−
x
)
(
x
−
x
)
p
(
x
|
y
)
d
x
.
(B.9)
Taking the derivative of the above integral with respect to
x
, and setting it to zero,
we have
∞
2
∞
−∞
∂
∂
T
(
x
−
x
)
(
x
−
x
)
p
(
x
|
y
)
d
x
=
(
x
−
x
)
p
(
x
|
y
)
d
x
=
0
.
(B.10)
x
−∞
Therefore, the MMSE estimate is equal to
∞
x
=
x
p
(
x
|
y
)
d
x
.
(B.11)
−∞
That is, the MMSE estimate is equal to the mean of the posterior. Note that, when
the posterior distribution is Gaussian, the MAP estimate and the MMSE estimate are
equal, because the Gaussian distribution reaches its maximum at the mean.
B.3 Derivation of Posterior Distribution
in the Gaussian Model
Let us consider the problem in which the time series of unknown parameters
x
1
,
x
2
,...,
x
K
are estimated using the time series of observation data
y
1
,
y
2
,...,
y
K
where
y
(
t
k
)
and
x
(
t
k
)
are denoted
y
k
and
x
k
. The relationship in Eq. (
B.1
) holds
between
x
k
and
y
k
, i.e.,
y
k
=
Hx
k
+
ʵ
.
(B.12)
