<s>
In	O
estimation	O
theory	O
and	O
decision	O
theory	O
,	O
a	O
Bayes	O
estimator	O
or	O
a	O
Bayes	B-General_Concept
action	I-General_Concept
is	O
an	O
estimator	O
or	O
decision	O
rule	O
that	O
minimizes	O
the	O
posterior	O
expected	O
value	O
of	O
a	O
loss	O
function	O
(	O
i.e.	O
,	O
the	O
posterior	O
expected	O
loss	O
)	O
.	O
</s>
<s>
An	O
alternative	O
way	O
of	O
formulating	O
an	O
estimator	O
within	O
Bayesian	O
statistics	O
is	O
maximum	B-General_Concept
a	I-General_Concept
posteriori	I-General_Concept
estimation	I-General_Concept
.	O
</s>
<s>
The	O
most	O
common	O
risk	O
function	O
used	O
for	O
Bayesian	O
estimation	O
is	O
the	O
mean	B-Algorithm
square	I-Algorithm
error	I-Algorithm
(	O
MSE	O
)	O
,	O
also	O
called	O
squared	O
error	O
risk	O
.	O
</s>
<s>
This	O
is	O
known	O
as	O
the	O
minimum	O
mean	B-Algorithm
square	I-Algorithm
error	I-Algorithm
(	O
MMSE	O
)	O
estimator	O
.	O
</s>
<s>
A	O
conjugate	O
prior	O
is	O
defined	O
as	O
a	O
prior	O
distribution	O
belonging	O
to	O
some	O
parametric	B-General_Concept
family	O
,	O
for	O
which	O
the	O
resulting	O
posterior	O
distribution	O
also	O
belongs	O
to	O
the	O
same	O
family	O
.	O
</s>
<s>
The	O
following	O
loss	O
function	O
is	O
trickier	O
:	O
it	O
yields	O
either	O
the	O
posterior	B-General_Concept
mode	I-General_Concept
,	O
or	O
a	O
point	O
close	O
to	O
it	O
depending	O
on	O
the	O
curvature	O
and	O
properties	O
of	O
the	O
posterior	O
distribution	O
.	O
</s>
<s>
Other	O
loss	O
functions	O
can	O
be	O
conceived	O
,	O
although	O
the	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
is	O
the	O
most	O
widely	O
used	O
and	O
validated	O
.	O
</s>
<s>
A	O
Bayes	O
estimator	O
derived	O
through	O
the	O
empirical	B-General_Concept
Bayes	I-General_Concept
method	I-General_Concept
is	O
called	O
an	O
empirical	B-General_Concept
Bayes	I-General_Concept
estimator	I-General_Concept
.	O
</s>
<s>
Empirical	B-General_Concept
Bayes	I-General_Concept
methods	I-General_Concept
enable	O
the	O
use	O
of	O
auxiliary	O
empirical	O
data	O
,	O
from	O
observations	O
of	O
related	O
parameters	O
,	O
in	O
the	O
development	O
of	O
a	O
Bayes	O
estimator	O
.	O
</s>
<s>
There	O
are	O
parametric	B-General_Concept
and	O
non-parametric	B-General_Concept
approaches	O
to	O
empirical	B-General_Concept
Bayes	I-General_Concept
estimation	O
.	O
</s>
<s>
Parametric	B-General_Concept
empirical	B-General_Concept
Bayes	I-General_Concept
is	O
usually	O
preferable	O
since	O
it	O
is	O
more	O
applicable	O
and	O
more	O
accurate	O
on	O
small	O
amounts	O
of	O
data	O
.	O
</s>
<s>
The	O
following	O
is	O
a	O
simple	O
example	O
of	O
parametric	B-General_Concept
empirical	B-General_Concept
Bayes	I-General_Concept
estimation	O
.	O
</s>
<s>
For	O
example	O
,	O
as	O
stated	O
above	O
,	O
under	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
(	O
MSE	O
)	O
the	O
Bayes	O
rule	O
is	O
unique	O
and	O
therefore	O
admissible	O
.	O
</s>
