<s>
In	O
statistics	O
and	O
signal	O
processing	O
,	O
a	O
minimum	B-General_Concept
mean	I-General_Concept
square	I-General_Concept
error	I-General_Concept
(	O
MMSE	O
)	O
estimator	O
is	O
an	O
estimation	O
method	O
which	O
minimizes	O
the	O
mean	B-Algorithm
square	I-Algorithm
error	I-Algorithm
(	O
MSE	O
)	O
,	O
which	O
is	O
a	O
common	O
measure	O
of	O
estimator	O
quality	O
,	O
of	O
the	O
fitted	O
values	O
of	O
a	O
dependent	O
variable	O
.	O
</s>
<s>
In	O
the	O
Bayesian	B-General_Concept
setting	O
,	O
the	O
term	O
MMSE	O
more	O
specifically	O
refers	O
to	O
estimation	O
with	O
quadratic	O
loss	O
function	O
.	O
</s>
<s>
In	O
such	O
case	O
,	O
the	O
MMSE	B-General_Concept
estimator	I-General_Concept
is	O
given	O
by	O
the	O
posterior	O
mean	O
of	O
the	O
parameter	O
to	O
be	O
estimated	O
.	O
</s>
<s>
Since	O
the	O
posterior	O
mean	O
is	O
cumbersome	O
to	O
calculate	O
,	O
the	O
form	O
of	O
the	O
MMSE	B-General_Concept
estimator	I-General_Concept
is	O
usually	O
constrained	O
to	O
be	O
within	O
a	O
certain	O
class	O
of	O
functions	O
.	O
</s>
<s>
Linear	O
MMSE	B-General_Concept
estimators	I-General_Concept
are	O
a	O
popular	O
choice	O
since	O
they	O
are	O
easy	O
to	O
use	O
,	O
easy	O
to	O
calculate	O
,	O
and	O
very	O
versatile	O
.	O
</s>
<s>
The	O
term	O
MMSE	O
more	O
specifically	O
refers	O
to	O
estimation	O
in	O
a	O
Bayesian	B-General_Concept
setting	O
with	O
quadratic	O
cost	O
function	O
.	O
</s>
<s>
The	O
basic	O
idea	O
behind	O
the	O
Bayesian	B-General_Concept
approach	O
to	O
estimation	O
stems	O
from	O
practical	O
situations	O
where	O
we	O
often	O
have	O
some	O
prior	O
information	O
about	O
the	O
parameter	O
to	O
be	O
estimated	O
.	O
</s>
<s>
This	O
is	O
in	O
contrast	O
to	O
the	O
non-Bayesian	O
approach	O
like	O
minimum-variance	O
unbiased	O
estimator	O
(	O
MVUE	O
)	O
where	O
absolutely	O
nothing	O
is	O
assumed	O
to	O
be	O
known	O
about	O
the	O
parameter	O
in	O
advance	O
and	O
which	O
does	O
not	O
account	O
for	O
such	O
situations	O
.	O
</s>
<s>
In	O
the	O
Bayesian	B-General_Concept
approach	O
,	O
such	O
prior	O
information	O
is	O
captured	O
by	O
the	O
prior	O
probability	O
density	O
function	O
of	O
the	O
parameters	O
;	O
and	O
based	O
directly	O
on	O
Bayes	O
theorem	O
,	O
it	O
allows	O
us	O
to	O
make	O
better	O
posterior	O
estimates	O
as	O
more	O
observations	O
become	O
available	O
.	O
</s>
<s>
Thus	O
unlike	O
non-Bayesian	O
approach	O
where	O
parameters	O
of	O
interest	O
are	O
assumed	O
to	O
be	O
deterministic	O
,	O
but	O
unknown	O
constants	O
,	O
the	O
Bayesian	B-General_Concept
estimator	I-General_Concept
seeks	O
to	O
estimate	O
a	O
parameter	O
that	O
is	O
itself	O
a	O
random	O
variable	O
.	O
</s>
<s>
Furthermore	O
,	O
Bayesian	B-General_Concept
estimation	I-General_Concept
can	O
also	O
deal	O
with	O
situations	O
where	O
the	O
sequence	O
of	O
observations	O
are	O
not	O
necessarily	O
independent	O
.	O
</s>
<s>
Thus	O
Bayesian	B-General_Concept
estimation	I-General_Concept
provides	O
yet	O
another	O
alternative	O
to	O
the	O
MVUE	O
.	O
</s>
<s>
The	O
MMSE	B-General_Concept
estimator	I-General_Concept
is	O
then	O
defined	O
as	O
the	O
estimator	O
achieving	O
minimal	O
MSE	O
:	O
</s>
<s>
When	O
the	O
means	O
and	O
variances	O
are	O
finite	O
,	O
the	O
MMSE	B-General_Concept
estimator	I-General_Concept
is	O
uniquely	O
defined	O
and	O
is	O
given	O
by	O
:	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
MMSE	B-General_Concept
estimator	I-General_Concept
is	O
the	O
conditional	O
expectation	O
of	O
given	O
the	O
known	O
observed	O
value	O
of	O
the	O
measurements	O
.	O
</s>
<s>
The	O
MMSE	B-General_Concept
estimator	I-General_Concept
is	O
unbiased	O
(	O
under	O
the	O
regularity	O
assumptions	O
mentioned	O
above	O
)	O
:	O
</s>
<s>
The	O
MMSE	B-General_Concept
estimator	I-General_Concept
is	O
asymptotically	O
unbiased	O
and	O
it	O
converges	O
in	O
distribution	O
to	O
the	O
normal	O
distribution	O
:	O
</s>
<s>
Thus	O
,	O
the	O
MMSE	B-General_Concept
estimator	I-General_Concept
is	O
asymptotically	O
efficient	O
.	O
</s>
<s>
The	O
orthogonality	B-General_Concept
principle	I-General_Concept
:	O
When	O
is	O
a	O
scalar	O
,	O
an	O
estimator	O
constrained	O
to	O
be	O
of	O
certain	O
form	O
is	O
an	O
optimal	O
estimator	O
,	O
i.e.	O
</s>
<s>
For	O
random	O
vectors	O
,	O
since	O
the	O
MSE	O
for	O
estimation	O
of	O
a	O
random	O
vector	O
is	O
the	O
sum	O
of	O
the	O
MSEs	O
of	O
the	O
coordinates	O
,	O
finding	O
the	O
MMSE	B-General_Concept
estimator	I-General_Concept
of	O
a	O
random	O
vector	O
decomposes	O
into	O
finding	O
the	O
MMSE	B-General_Concept
estimators	I-General_Concept
of	O
the	O
coordinates	O
of	O
X	O
separately	O
:	O
</s>
<s>
If	O
and	O
are	O
jointly	O
Gaussian	O
,	O
then	O
the	O
MMSE	B-General_Concept
estimator	I-General_Concept
is	O
linear	O
,	O
i.e.	O
,	O
it	O
has	O
the	O
form	O
for	O
matrix	O
and	O
constant	O
.	O
</s>
<s>
As	O
a	O
consequence	O
,	O
to	O
find	O
the	O
MMSE	B-General_Concept
estimator	I-General_Concept
,	O
it	O
is	O
sufficient	O
to	O
find	O
the	O
linear	O
MMSE	B-General_Concept
estimator	I-General_Concept
.	O
</s>
<s>
In	O
many	O
cases	O
,	O
it	O
is	O
not	O
possible	O
to	O
determine	O
the	O
analytical	O
expression	O
of	O
the	O
MMSE	B-General_Concept
estimator	I-General_Concept
.	O
</s>
<s>
Direct	O
numerical	O
evaluation	O
of	O
the	O
conditional	O
expectation	O
is	O
computationally	O
expensive	O
since	O
it	O
often	O
requires	O
multidimensional	O
integration	O
usually	O
done	O
via	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Another	O
computational	O
approach	O
is	O
to	O
directly	O
seek	O
the	O
minima	O
of	O
the	O
MSE	O
using	O
techniques	O
such	O
as	O
the	O
stochastic	B-Algorithm
gradient	I-Algorithm
descent	I-Algorithm
methods	I-Algorithm
;	O
but	O
this	O
method	O
still	O
requires	O
the	O
evaluation	O
of	O
expectation	O
.	O
</s>
<s>
While	O
these	O
numerical	O
methods	O
have	O
been	O
fruitful	O
,	O
a	O
closed	O
form	O
expression	O
for	O
the	O
MMSE	B-General_Concept
estimator	I-General_Concept
is	O
nevertheless	O
possible	O
if	O
we	O
are	O
willing	O
to	O
make	O
some	O
compromises	O
.	O
</s>
<s>
The	O
linear	O
MMSE	B-General_Concept
estimator	I-General_Concept
is	O
the	O
estimator	O
achieving	O
minimum	O
MSE	O
among	O
all	O
estimators	O
of	O
such	O
form	O
.	O
</s>
<s>
One	O
advantage	O
of	O
such	O
linear	O
MMSE	B-General_Concept
estimator	I-General_Concept
is	O
that	O
it	O
is	O
not	O
necessary	O
to	O
explicitly	O
calculate	O
the	O
posterior	O
probability	O
density	O
function	O
of	O
.	O
</s>
<s>
Let	O
us	O
have	O
the	O
optimal	O
linear	O
MMSE	B-General_Concept
estimator	I-General_Concept
given	O
as	O
,	O
where	O
we	O
are	O
required	O
to	O
find	O
the	O
expression	O
for	O
and	O
.	O
</s>
<s>
It	O
is	O
required	O
that	O
the	O
MMSE	B-General_Concept
estimator	I-General_Concept
be	O
unbiased	O
.	O
</s>
<s>
From	O
the	O
orthogonality	B-General_Concept
principle	I-General_Concept
,	O
we	O
can	O
have	O
,	O
where	O
we	O
take	O
.	O
</s>
<s>
The	O
first	O
term	O
in	O
the	O
third	O
line	O
is	O
zero	O
due	O
to	O
the	O
orthogonality	B-General_Concept
principle	I-General_Concept
.	O
</s>
<s>
Standard	O
method	O
like	O
Gauss	B-Algorithm
elimination	I-Algorithm
can	O
be	O
used	O
to	O
solve	O
the	O
matrix	O
equation	O
for	O
.	O
</s>
<s>
Since	O
the	O
matrix	O
is	O
a	O
symmetric	O
positive	O
definite	O
matrix	O
,	O
can	O
be	O
solved	O
twice	O
as	O
fast	O
with	O
the	O
Cholesky	O
decomposition	O
,	O
while	O
for	O
large	O
sparse	O
systems	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
is	O
more	O
effective	O
.	O
</s>
<s>
Levinson	B-Algorithm
recursion	I-Algorithm
is	O
a	O
fast	O
method	O
when	O
is	O
also	O
a	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
This	O
can	O
happen	O
when	O
is	O
a	O
wide	B-Algorithm
sense	I-Algorithm
stationary	I-Algorithm
process	O
.	O
</s>
<s>
The	O
significant	O
difference	O
between	O
the	O
estimation	O
problem	O
treated	O
above	O
and	O
those	O
of	O
least	B-Algorithm
squares	I-Algorithm
and	O
Gauss	O
–	O
Markov	O
estimate	O
is	O
that	O
the	O
number	O
of	O
observations	O
m	O
,	O
(	O
i.e.	O
</s>
<s>
In	O
the	O
Bayesian	B-General_Concept
framework	O
,	O
such	O
recursive	O
estimation	O
is	O
easily	O
facilitated	O
using	O
Bayes	O
 '	O
rule	O
.	O
</s>
<s>
With	O
the	O
lack	O
of	O
dynamical	O
information	O
on	O
how	O
the	O
state	O
changes	O
with	O
time	O
,	O
we	O
will	O
make	O
a	O
further	O
stationarity	B-Algorithm
assumption	O
about	O
the	O
prior	O
:	O
</s>
<s>
In	O
the	O
context	O
of	O
linear	O
MMSE	B-General_Concept
estimator	I-General_Concept
,	O
the	O
formula	O
for	O
the	O
estimate	O
will	O
have	O
the	O
same	O
form	O
as	O
before	O
:	O
However	O
,	O
the	O
mean	O
and	O
covariance	O
matrices	O
of	O
and	O
will	O
need	O
to	O
be	O
replaced	O
by	O
those	O
of	O
the	O
prior	O
density	O
and	O
likelihood	O
,	O
respectively	O
.	O
</s>
<s>
as	O
per	O
by	O
the	O
properties	O
of	O
MMSE	B-General_Concept
estimators	I-General_Concept
and	O
the	O
stationarity	B-Algorithm
assumption	O
.	O
</s>
<s>
The	O
generalization	O
of	O
this	O
idea	O
to	O
non-stationary	B-Algorithm
cases	O
gives	O
rise	O
to	O
the	O
Kalman	O
filter	O
.	O
</s>
<s>
Alternative	O
approaches	O
:	O
This	O
important	O
special	O
case	O
has	O
also	O
given	O
rise	O
to	O
many	O
other	O
iterative	O
methods	O
(	O
or	O
adaptive	O
filters	O
)	O
,	O
such	O
as	O
the	O
least	O
mean	O
squares	O
filter	O
and	O
recursive	O
least	B-Algorithm
squares	I-Algorithm
filter	O
,	O
that	O
directly	O
solves	O
the	O
original	O
MSE	O
optimization	O
problem	O
using	O
stochastic	B-Algorithm
gradient	I-Algorithm
descents	I-Algorithm
.	O
</s>
<s>
mean	B-Algorithm
square	I-Algorithm
error	I-Algorithm
then	O
gives	O
.	O
</s>
<s>
The	O
matrix	O
equation	O
can	O
be	O
solved	O
by	O
well	O
known	O
methods	O
such	O
as	O
Gauss	B-Algorithm
elimination	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
A	O
shorter	O
,	O
non-numerical	O
example	O
can	O
be	O
found	O
in	O
orthogonality	B-General_Concept
principle	I-General_Concept
.	O
</s>
<s>
while	O
the	O
variance	O
will	O
be	O
unaffected	O
by	O
data	O
and	O
the	O
LMMSE	B-General_Concept
of	O
the	O
estimate	O
will	O
tend	O
to	O
zero	O
.	O
</s>
