<s>
In	O
estimation	O
theory	O
and	O
decision	O
theory	O
,	O
a	O
Bayes	B-General_Concept
estimator	I-General_Concept
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
Bayes	B-General_Concept
risk	I-General_Concept
of	O
is	O
defined	O
as	O
,	O
where	O
the	O
expectation	O
is	O
taken	O
over	O
the	O
probability	O
distribution	O
of	O
:	O
this	O
defines	O
the	O
risk	O
function	O
as	O
a	O
function	O
of	O
.	O
</s>
<s>
An	O
estimator	O
is	O
said	O
to	O
be	O
a	O
Bayes	B-General_Concept
estimator	I-General_Concept
if	O
it	O
minimizes	O
the	O
Bayes	B-General_Concept
risk	I-General_Concept
among	O
all	O
estimators	O
.	O
</s>
<s>
Equivalently	O
,	O
the	O
estimator	O
which	O
minimizes	O
the	O
posterior	O
expected	O
loss	O
for	O
each	O
also	O
minimizes	O
the	O
Bayes	B-General_Concept
risk	I-General_Concept
and	O
therefore	O
is	O
a	O
Bayes	B-General_Concept
estimator	I-General_Concept
.	O
</s>
<s>
If	O
the	O
prior	O
is	O
improper	O
then	O
an	O
estimator	O
which	O
minimizes	O
the	O
posterior	O
expected	O
loss	O
for	O
each	O
is	O
called	O
a	O
generalized	O
Bayes	B-General_Concept
estimator	I-General_Concept
.	O
</s>
<s>
The	O
most	O
common	O
risk	O
function	O
used	O
for	O
Bayesian	B-General_Concept
estimation	I-General_Concept
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>
This	O
is	O
an	O
important	O
property	O
,	O
since	O
the	O
Bayes	B-General_Concept
estimator	I-General_Concept
,	O
as	O
well	O
as	O
its	O
statistical	O
properties	O
(	O
variance	O
,	O
confidence	O
interval	O
,	O
etc	O
.	O
</s>
<s>
In	O
sequential	O
estimation	O
,	O
unless	O
a	O
conjugate	O
prior	O
is	O
used	O
,	O
the	O
posterior	O
distribution	O
typically	O
becomes	O
more	O
complex	O
with	O
each	O
added	O
measurement	O
,	O
and	O
the	O
Bayes	B-General_Concept
estimator	I-General_Concept
cannot	O
usually	O
be	O
calculated	O
without	O
resorting	O
to	O
numerical	O
methods	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>
The	O
use	O
of	O
an	O
improper	O
prior	O
means	O
that	O
the	O
Bayes	B-General_Concept
risk	I-General_Concept
is	O
undefined	O
(	O
since	O
the	O
prior	O
is	O
not	O
a	O
probability	O
distribution	O
and	O
we	O
cannot	O
take	O
an	O
expectation	O
under	O
it	O
)	O
.	O
</s>
<s>
As	O
a	O
consequence	O
,	O
it	O
is	O
no	O
longer	O
meaningful	O
to	O
speak	O
of	O
a	O
Bayes	B-General_Concept
estimator	I-General_Concept
that	O
minimizes	O
the	O
Bayes	B-General_Concept
risk	I-General_Concept
.	O
</s>
<s>
Recall	O
that	O
,	O
for	O
a	O
proper	O
prior	O
,	O
the	O
Bayes	B-General_Concept
estimator	I-General_Concept
minimizes	O
the	O
posterior	O
expected	O
loss	O
.	O
</s>
<s>
When	O
the	O
prior	O
is	O
improper	O
,	O
an	O
estimator	O
which	O
minimizes	O
the	O
posterior	O
expected	O
loss	O
is	O
referred	O
to	O
as	O
a	O
generalized	O
Bayes	B-General_Concept
estimator	I-General_Concept
.	O
</s>
<s>
The	O
generalized	O
Bayes	B-General_Concept
estimator	I-General_Concept
is	O
the	O
value	O
that	O
minimizes	O
this	O
expression	O
for	O
a	O
given	O
.	O
</s>
<s>
In	O
this	O
case	O
it	O
can	O
be	O
shown	O
that	O
the	O
generalized	O
Bayes	B-General_Concept
estimator	I-General_Concept
has	O
the	O
form	O
,	O
for	O
some	O
constant	O
.	O
</s>
<s>
A	O
Bayes	B-General_Concept
estimator	I-General_Concept
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	B-General_Concept
estimator	I-General_Concept
.	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
if	O
,	O
and	O
if	O
we	O
assume	O
a	O
normal	O
prior	O
(	O
which	O
is	O
a	O
conjugate	O
prior	O
in	O
this	O
case	O
)	O
,	O
we	O
conclude	O
that	O
,	O
from	O
which	O
the	O
Bayes	B-General_Concept
estimator	I-General_Concept
of	O
based	O
on	O
can	O
be	O
calculated	O
.	O
</s>
<s>
Bayes	O
rules	O
having	O
finite	O
Bayes	B-General_Concept
risk	I-General_Concept
are	O
typically	O
admissible	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>
<s>
By	O
contrast	O
,	O
generalized	O
Bayes	O
rules	O
often	O
have	O
undefined	O
Bayes	B-General_Concept
risk	I-General_Concept
in	O
the	O
case	O
of	O
improper	O
priors	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
generalized	O
Bayes	B-General_Concept
estimator	I-General_Concept
of	O
a	O
location	O
parameter	O
θ	O
based	O
on	O
Gaussian	O
samples	O
(	O
described	O
in	O
the	O
"	O
Generalized	O
Bayes	B-General_Concept
estimator	I-General_Concept
"	O
section	O
above	O
)	O
is	O
inadmissible	O
for	O
;	O
this	O
is	O
known	O
as	O
Stein	O
's	O
phenomenon	O
.	O
</s>
<s>
Let	O
be	O
a	O
sequence	O
of	O
Bayes	B-General_Concept
estimators	I-General_Concept
of	O
θ	O
based	O
on	O
an	O
increasing	O
number	O
of	O
measurements	O
.	O
</s>
<s>
Moreover	O
,	O
if	O
δ	O
is	O
the	O
Bayes	B-General_Concept
estimator	I-General_Concept
under	O
MSE	O
risk	O
,	O
then	O
it	O
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>
It	O
follows	O
that	O
the	O
Bayes	B-General_Concept
estimator	I-General_Concept
δn	O
under	O
MSE	O
is	O
asymptotically	O
efficient	O
.	O
</s>
<s>
The	O
relations	O
between	O
the	O
maximum	O
likelihood	O
and	O
Bayes	B-General_Concept
estimators	I-General_Concept
can	O
be	O
shown	O
in	O
the	O
following	O
simple	O
example	O
.	O
</s>
<s>
The	O
last	O
equation	O
implies	O
that	O
,	O
for	O
n	O
→	O
∞	O
,	O
the	O
Bayes	B-General_Concept
estimator	I-General_Concept
(	O
in	O
the	O
described	O
problem	O
)	O
is	O
close	O
to	O
the	O
MLE	O
.	O
</s>
<s>
The	O
Internet	O
Movie	O
Database	O
uses	O
a	O
formula	O
for	O
calculating	O
and	O
comparing	O
the	O
ratings	O
of	O
films	O
by	O
its	O
users	O
,	O
including	O
their	O
Top	O
Rated	O
250	O
Titles	O
which	O
is	O
claimed	O
to	O
give	O
"	O
a	O
true	O
Bayesian	B-General_Concept
estimate	I-General_Concept
"	O
.	O
</s>
