<s>
In	O
applied	O
statistics	O
,	O
optimal	B-General_Concept
estimation	I-General_Concept
is	O
a	O
regularized	O
matrix	B-Architecture
inverse	O
method	O
based	O
on	O
Bayes	O
 '	O
theorem	O
.	O
</s>
<s>
A	O
matrix	B-Architecture
inverse	O
problem	O
looks	O
like	O
this	O
:	O
</s>
<s>
The	O
essential	O
concept	O
is	O
to	O
transform	O
the	O
matrix	B-Architecture
,	O
A	O
,	O
into	O
a	O
conditional	O
probability	O
and	O
the	O
variables	O
,	O
and	O
into	O
probability	O
distributions	O
by	O
assuming	O
Gaussian	O
statistics	O
and	O
using	O
empirically-determined	O
covariance	O
matrices	O
.	O
</s>
<s>
where	O
m	O
and	O
n	O
are	O
the	O
numbers	O
of	O
elements	O
in	O
and	O
respectively	O
is	O
the	O
matrix	B-Architecture
to	O
be	O
solved	O
(	O
the	O
linear	O
or	O
linearised	O
forward	O
model	O
)	O
and	O
is	O
the	O
covariance	O
matrix	B-Architecture
of	O
the	O
vector	O
.	O
</s>
<s>
Here	O
is	O
taken	O
to	O
be	O
the	O
so-called	O
"	O
a-priori	O
"	O
distribution	O
:	O
denotes	O
the	O
a-priori	O
values	O
for	O
while	O
is	O
its	O
covariance	O
matrix	B-Architecture
.	O
</s>
<s>
Now	O
it	O
is	O
possible	O
to	O
solve	O
for	O
both	O
the	O
expectation	O
value	O
of	O
,	O
,	O
and	O
for	O
its	O
covariance	O
matrix	B-Architecture
by	O
equating	O
and	O
.	O
</s>
<s>
Typically	O
with	O
optimal	B-General_Concept
estimation	I-General_Concept
,	O
in	O
addition	O
to	O
the	O
vector	O
of	O
retrieved	O
quantities	O
,	O
one	O
extra	O
matrix	B-Architecture
is	O
returned	O
along	O
with	O
the	O
covariance	O
matrix	B-Architecture
.	O
</s>
<s>
This	O
is	O
sometimes	O
called	O
the	O
resolution	O
matrix	B-Architecture
or	O
the	O
averaging	O
kernel	O
and	O
is	O
calculated	O
as	O
follows	O
:	O
</s>
