<s>
In	O
statistics	O
,	O
the	O
precision	B-General_Concept
matrix	I-General_Concept
or	O
concentration	B-General_Concept
matrix	I-General_Concept
is	O
the	O
matrix	O
inverse	O
of	O
the	O
covariance	O
matrix	O
or	O
dispersion	O
matrix	O
,	O
.	O
</s>
<s>
For	O
univariate	O
distributions	O
,	O
the	O
precision	B-General_Concept
matrix	I-General_Concept
degenerates	O
into	O
a	O
scalar	O
precision	B-General_Concept
,	O
defined	O
as	O
the	O
reciprocal	O
of	O
the	O
variance	O
,	O
.	O
</s>
<s>
include	O
the	O
reciprocal	O
of	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
;	O
</s>
<s>
the	O
standard	B-General_Concept
deviation	I-General_Concept
itself	O
and	O
the	O
relative	O
standard	B-General_Concept
deviation	I-General_Concept
;	O
</s>
<s>
as	O
well	O
as	O
the	O
standard	B-General_Concept
error	I-General_Concept
and	O
the	O
confidence	O
interval	O
(	O
or	O
its	O
half-width	O
,	O
the	O
margin	O
of	O
error	O
)	O
.	O
</s>
<s>
One	O
particular	O
use	O
of	O
the	O
precision	B-General_Concept
matrix	I-General_Concept
is	O
in	O
the	O
context	O
of	O
Bayesian	O
analysis	O
of	O
the	O
multivariate	O
normal	O
distribution	O
:	O
for	O
example	O
,	O
Bernardo	O
&	O
Smith	O
prefer	O
to	O
parameterise	O
the	O
multivariate	O
normal	O
distribution	O
in	O
terms	O
of	O
the	O
precision	B-General_Concept
matrix	I-General_Concept
,	O
rather	O
than	O
the	O
covariance	O
matrix	O
,	O
because	O
of	O
certain	O
simplifications	O
that	O
then	O
arise	O
.	O
</s>
<s>
For	O
instance	O
,	O
if	O
both	O
the	O
prior	O
and	O
the	O
likelihood	O
have	O
Gaussian	O
form	O
,	O
and	O
the	O
precision	B-General_Concept
matrix	I-General_Concept
of	O
both	O
of	O
these	O
exist	O
(	O
because	O
their	O
covariance	O
matrix	O
is	O
full	O
rank	O
and	O
thus	O
invertible	O
)	O
,	O
then	O
the	O
precision	B-General_Concept
matrix	I-General_Concept
of	O
the	O
posterior	O
will	O
simply	O
be	O
the	O
sum	O
of	O
the	O
precision	B-General_Concept
matrices	O
of	O
the	O
prior	O
and	O
the	O
likelihood	O
.	O
</s>
<s>
As	O
the	O
inverse	O
of	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
,	O
the	O
precision	B-General_Concept
matrix	I-General_Concept
of	O
real-valued	O
random	O
variables	O
,	O
if	O
it	O
exists	O
,	O
is	O
positive	B-Algorithm
definite	I-Algorithm
and	O
symmetrical	O
.	O
</s>
<s>
Another	O
reason	O
the	O
precision	B-General_Concept
matrix	I-General_Concept
may	O
be	O
useful	O
is	O
that	O
if	O
two	O
dimensions	O
and	O
of	O
a	O
multivariate	O
normal	O
are	O
conditionally	O
independent	O
,	O
then	O
the	O
and	O
elements	O
of	O
the	O
precision	B-General_Concept
matrix	I-General_Concept
are	O
.	O
</s>
<s>
This	O
means	O
that	O
precision	B-General_Concept
matrices	O
tend	O
to	O
be	O
sparse	O
when	O
many	O
of	O
the	O
dimensions	O
are	O
conditionally	O
independent	O
,	O
which	O
can	O
lead	O
to	O
computational	O
efficiencies	O
when	O
working	O
with	O
them	O
.	O
</s>
<s>
It	O
also	O
means	O
that	O
precision	B-General_Concept
matrices	O
are	O
closely	O
related	O
to	O
the	O
idea	O
of	O
partial	B-Language
correlation	I-Language
.	O
</s>
<s>
The	O
precision	B-General_Concept
matrix	I-General_Concept
plays	O
a	O
central	O
role	O
in	O
generalized	O
least	O
squares	O
,	O
compared	O
to	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
,	O
where	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
,	O
and	O
to	O
weighted	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
,	O
where	O
is	O
diagonal	O
(	O
the	O
weight	O
matrix	O
)	O
.	O
</s>
<s>
The	O
term	O
precision	B-General_Concept
in	O
this	O
sense	O
(	O
“	O
mensura	O
praecisionis	O
observationum	O
”	O
)	O
first	O
appeared	O
in	O
the	O
works	O
of	O
Gauss	O
(	O
1809	O
)	O
“	O
Theoria	O
motus	O
corporum	O
coelestium	O
in	O
sectionibus	O
conicis	O
solem	O
ambientium	O
”	O
(	O
page	O
212	O
)	O
.	O
</s>
<s>
He	O
writes	O
,	O
for	O
the	O
density	O
function	O
of	O
a	O
normal	O
distribution	O
with	O
precision	B-General_Concept
(	O
reciprocal	O
of	O
standard	B-General_Concept
deviation	I-General_Concept
)	O
,	O
</s>
<s>
Later	O
Whittaker	O
&	O
Robinson	O
(	O
1924	O
)	O
“	O
Calculus	O
of	O
observations	O
”	O
called	O
this	O
quantity	O
the	O
modulus	O
(	O
of	O
precision	B-General_Concept
)	O
,	O
but	O
this	O
term	O
has	O
dropped	O
out	O
of	O
use	O
.	O
</s>
