<s>
In	O
Bayesian	O
statistics	O
,	O
a	O
maximum	B-General_Concept
a	I-General_Concept
posteriori	I-General_Concept
probability	O
(	O
MAP	O
)	O
estimate	O
is	O
an	O
estimate	O
of	O
an	O
unknown	O
quantity	O
,	O
that	O
equals	O
the	O
mode	O
of	O
the	O
posterior	O
distribution	O
.	O
</s>
<s>
MAP	B-General_Concept
estimation	I-General_Concept
can	O
therefore	O
be	O
seen	O
as	O
a	O
regularization	O
of	O
maximum	O
likelihood	O
estimation	O
.	O
</s>
<s>
Let	O
be	O
the	O
sampling	B-General_Concept
distribution	I-General_Concept
of	O
,	O
so	O
that	O
is	O
the	O
probability	O
of	O
when	O
the	O
underlying	O
population	O
parameter	O
is	O
.	O
</s>
<s>
The	O
method	O
of	O
maximum	B-General_Concept
a	I-General_Concept
posteriori	I-General_Concept
estimation	I-General_Concept
then	O
estimates	O
as	O
the	O
mode	O
of	O
the	O
posterior	O
distribution	O
of	O
this	O
random	O
variable	O
:	O
</s>
<s>
Observe	O
that	O
the	O
MAP	B-General_Concept
estimate	I-General_Concept
of	O
coincides	O
with	O
the	O
ML	O
estimate	O
when	O
the	O
prior	O
is	O
uniform	O
(	O
i.e.	O
,	O
is	O
a	O
constant	O
function	O
)	O
.	O
</s>
<s>
as	O
goes	O
to	O
0	O
,	O
the	O
Bayes	B-General_Concept
estimator	I-General_Concept
approaches	O
the	O
MAP	B-General_Concept
estimator	I-General_Concept
,	O
provided	O
that	O
the	O
distribution	O
of	O
is	O
quasi-concave	O
.	O
</s>
<s>
But	O
generally	O
a	O
MAP	B-General_Concept
estimator	I-General_Concept
is	O
not	O
a	O
Bayes	B-General_Concept
estimator	I-General_Concept
unless	O
is	O
discrete	O
.	O
</s>
<s>
MAP	B-General_Concept
estimates	I-General_Concept
can	O
be	O
computed	O
in	O
several	O
ways	O
:	O
</s>
<s>
Via	O
numerical	B-General_Concept
optimization	O
such	O
as	O
the	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
or	O
Newton	B-Algorithm
's	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
This	O
usually	O
requires	O
first	O
or	O
second	O
derivatives	B-Algorithm
,	O
which	O
have	O
to	O
be	O
evaluated	O
analytically	O
or	O
numerically	B-General_Concept
.	O
</s>
<s>
Via	O
a	O
modification	O
of	O
an	O
expectation-maximization	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
This	O
does	O
not	O
require	O
derivatives	B-Algorithm
of	O
the	O
posterior	O
density	O
.	O
</s>
<s>
While	O
only	O
mild	O
conditions	O
are	O
required	O
for	O
MAP	B-General_Concept
estimation	I-General_Concept
to	O
be	O
a	O
limiting	O
case	O
of	O
Bayes	B-General_Concept
estimation	I-General_Concept
(	O
under	O
the	O
0	O
–	O
1	O
loss	O
function	O
)	O
,	O
it	O
is	O
not	O
very	O
representative	O
of	O
Bayesian	O
methods	O
in	O
general	O
.	O
</s>
<s>
This	O
is	O
because	O
MAP	B-General_Concept
estimates	I-General_Concept
are	O
point	O
estimates	O
,	O
whereas	O
Bayesian	O
methods	O
are	O
characterized	O
by	O
the	O
use	O
of	O
distributions	O
to	O
summarize	O
data	O
and	O
draw	O
inferences	O
:	O
thus	O
,	O
Bayesian	O
methods	O
tend	O
to	O
report	O
the	O
posterior	O
mean	O
or	O
median	O
instead	O
,	O
together	O
with	O
credible	B-General_Concept
intervals	I-General_Concept
.	O
</s>
<s>
In	O
addition	O
,	O
the	O
posterior	O
distribution	O
may	O
often	O
not	O
have	O
a	O
simple	O
analytic	O
form	O
:	O
in	O
this	O
case	O
,	O
the	O
distribution	O
can	O
be	O
simulated	O
using	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
techniques	O
,	O
while	O
optimization	O
to	O
find	O
its	O
mode(s )	O
may	O
be	O
difficult	O
or	O
impossible	O
.	O
</s>
<s>
Finally	O
,	O
unlike	O
ML	O
estimators	O
,	O
the	O
MAP	B-General_Concept
estimate	I-General_Concept
is	O
not	O
invariant	O
under	O
reparameterization	O
.	O
</s>
<s>
As	O
an	O
example	O
of	O
the	O
difference	O
between	O
Bayes	B-General_Concept
estimators	I-General_Concept
mentioned	O
above	O
(	O
mean	O
and	O
median	O
estimators	O
)	O
and	O
using	O
a	O
MAP	B-General_Concept
estimate	I-General_Concept
,	O
consider	O
the	O
case	O
where	O
there	O
is	O
a	O
need	O
to	O
classify	O
inputs	O
as	O
either	O
positive	O
or	O
negative	O
(	O
for	O
example	O
,	O
loans	O
as	O
risky	O
or	O
safe	O
)	O
.	O
</s>
<s>
Using	O
the	O
MAP	B-General_Concept
estimate	I-General_Concept
for	O
the	O
correct	O
classifier	O
,	O
is	O
classified	O
as	O
positive	O
,	O
whereas	O
the	O
Bayes	B-General_Concept
estimators	I-General_Concept
would	O
average	O
over	O
all	O
hypotheses	O
and	O
classify	O
as	O
negative	O
.	O
</s>
<s>
We	O
wish	O
to	O
find	O
the	O
MAP	B-General_Concept
estimate	I-General_Concept
of	O
.	O
</s>
