Databases Reference
In-Depth Information
|
−
x
)
g
H
(
y
)] with
g
(
y
)
=
x
−
|
The third term is of the form
E
[(
E
[
x
y
]
E
[
x
y
]. Using the
law of total expectation, we see that this term vanishes:
x
)
g
H
(
y
)
E
{
E
[(
E
[
x
|
y
]
−
|
y
]
}=
0
.
(5.7)
The same reasoning can be applied to conclude that the second term in (
5.6
) is also zero.
Therefore, the optimum estimator, obtained by making the fourth term in (
5.6
) equal to
zero, turns out to be the
conditional mean estimator
x
=
E
[
x
|
y
]. Let
e
=
x
−
x
=
E
[
x
|
y
]
−
x
(5.8)
E
[
x
]. This says that
x
is an
unbiased estimator
of
x
. The covariance matrix of the error vector is
=
0
, and thus
E
[
x
]
=
be the error vector. Its mean
E
[
e
]
E
[
ee
H
]
x
)
H
]
Q
=
=
E
[(
E
[
x
|
y
]
−
x
)(
E
[
x
|
y
]
−
.
(5.9)
E
[
e
e
H
]
Any competing estimator
x
with mean-squared error matrix
Q
=
=
E
[(
x
−
x
)(
x
−
x
)
H
] will be suboptimum in the sense that
Q
≥
Q
, meaning that
Q
−
Q
is
positive semidefinite. As a consequence, the conditional mean estimator is a minimum
mean-squared error (MMSE) estimator:
tr
Q
=
e
2
2
E
e
=
tr
Q
≤
E
.
(5.10)
But we can say more. For the class of real-valued
increasing
functions of matrices,
Q
Q
implies
f
(
Q
)
f
(
Q
). Thus we have the following result.
≤
≤
Result 5.1.
The conditional mean estimator E
[
x
y
]
minimizes (maximizes) any increas-
ing (decreasing) function of the mean-squared error matrix
Q
.
|
Besides the trace, the determinant is another example of an increasing function. Thus,
the volume of the error covariance ellipsoid is also minimized:
det
Q
.
det
Q
≤
(5.11)
A very important property of the MMSE estimator is the so-called
orthogonality
principle
, a special case of which we have already encountered in the derivation (
5.7
).
However, it holds much more generally.
Result 5.2.
The error vector
e
is orthogonal to every measurable function of
y
,
g
(
y
)
.
That is, E
[
eg
H
(
y
)]
=
E
{
(
E
[
x
|
y
]
−
x
)
g
H
(
y
)
}=
0
.
y
], this orthogonality condition says that the estimator error
e
is
orthogonal to the estimator
E
[
x
|
y
]. Moreover, since the conditional mean estimator
is an idempotent operator
Px
=
For
g
(
y
)
=
E
[
x
|
E
[
x
|
y
] - which is to say that
P
2
x
=
E
{
E
[
x
|
y
]
|
y
}=
=
Px
- we may think of the conditional mean estimator as a projection operator
that orthogonally resolves
x
into its estimator
E
[
x
E
[
x
|
y
]
|
y
] minus its estimator error
e
:
x
=
E
[
x
|
y
]
−
(
E
[
x
|
y
]
−
x
)
.
(5.12)
The conditional mean estimator summarizes everything of use for MMSE estimation
of
x
from
y
. This means conditioning
x
on
y
and
y
∗
would change nothing, since this has
already been done, so to speak, in the conditional mean
E
[
x
|
y
]. Similarly the conditional
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