In those cases in which the expected value of the a
posteriori density cannot be computed, a related but
simpler estimate, the maximum a posteriori
(MAP) estimate, can usually be evaluated. The estimate
θ̂MAPr
θ
MAP
r
equals the location of the maximum of the a
posteriori density. Assuming that this maximum can be
found by evaluating the derivative of the a
posteriori density, the MAP estimate is the solution
of the equation
∂∂θpθ|r|θ=θ̂MAP=0
θ
θ
MAP
θ
p
r
θ
0
(1)
Any scaling of the density by a positive quantity that depends
on
rr does not change
the location of the maximum. Symbolically,
p
θ
|
r
=
p
r
|
θ
p
θ
p
r
p
θ
|
r
p
r
|
θ
p
θ
p
r
; the derivative does not involve the denominator, and
this term can be ignored. Thus, the only quantities required to
compute
θ̂MAP
θ
MAP
are the likelihood function and the parameter's
a priori density.
Although not apparent in its definition, the MAP estimate does
satisfy an error criterion. Define a criterion that is zero over
a small range of values about
ε=0
ε
0
and a positive constant outside that
range. Minimization of the expected value of this criterion with
respect to
θ
̂
θ
is accomplished by centering the criterion function at
the maximum of the density. The region having the largest area
is thus "notched out," and the criterion is minimized. Whenever
the a posteriori density is symmetric and
unimodal, the MAP and MMSE estimates coincide. In Gaussian
problems, such as the last example, this equivalence is alway
valid. In more general circumstances, they differ.
Let the observations have the same form as the previous example,
but with the modification that the parameter is now uniformly
distributed over the interval
θ
1
θ
2
θ
1
θ
2
. The a posteriori mean cannot be
computed in closed form. To obtain the MAP estimate, we need
to find the location of the maximum of
∀θ,
θ
1
≤θ≤
θ
2
:pr|θpθ=1
θ
2
-
θ
1
∏l=0L-112π
σ
n
2ⅇ-12rl-θ
σ
n
2
θ
θ
1
θ
θ
2
p
θ
r
p
θ
1
θ
2
θ
1
l
0
L
1
1
2
σ
n
2
1
2
r
l
θ
σ
n
2
(2)
Evaluating the logarithm of this quantity does not change the
location of the maximum and simplifies the manipulations in
many problems. Here, the logarithm is
∀θ,
θ
1
≤θ≤
θ
2
:lnpr|θpθ=-ln
θ
2
-
θ
1
-∑l=0L-1rl-θ
σ
n
2+lnC
θ
θ
1
θ
θ
2
p
θ
r
p
θ
θ
2
θ
1
l
0
L
1
r
l
θ
σ
n
2
C
(3)
where
CC is a constant with
respect to
θθ. Assuming
that the maximum is interior to the domain of the parameter,
the MAP estimate is found to be the sample average
∑rlL
r
l
L
. If the average lies outside this interval, the
corresponding endpoint of the interval is the location of the
maximum. To summarize,
θ̂MAPr=
θ
1
if∑lrlL<
θ
1
∑lrlLif
θ
1
≤∑lrlL≤
θ
2
θ
2
if
θ
2
<∑lrlL
θ
MAP
r
θ
1
l
l
r
l
L
θ
1
l
l
r
l
L
θ
1
l
l
r
l
L
θ
2
θ
2
θ
2
l
l
r
l
L
(4)
The
a posteriori density is not symmetric
because of the finite domain of
θθ. Thus, the MAP estimate
is not equivalent to the MMSE estimate, and the accompanying
increase in the mean-squared error is difficult to
compute. When the sample average is the estimate, the estimate
is unbiased; otherwise it is biased. Asymptotically, the
variance of the average tends to zero, with the consequences
that the estimate is unbiased and consistent.
