Digital Signal Processing Reference
In-Depth Information
Thus, using Table 2.1 we get
⎧
⎨
⎩
0.18, for variety = Cabernet-Sauvignon
0, for variety = Tannat
0.11, for variety = Malbec
0.06, for variety = Merlot
P
(
variety, taste = plum
)
P
(
variety
)
=
.
Thus, in this case, even though the Malbec wine provides a highest prob-
ability of finding the flavor of plum, the MAP estimator changes the answer
to Cabernet-Sauvignon. This is because the taster now takes into account the
marginal probabilities
P
(
Cabernet
-
Sauvignon
)
0.2, which
increases the probability that the observed flavor of plum is originated by a
Cabernet-Sauvignon wine.
=
0.4 and
P
(
Malbec
)
=
2.5.4.2 Minimum Mean-Squared Error
As mentioned before, the estimation error is directly related to the efficiency
of the estimator. For multiple parameters, we can define the error vector
−
θ
=
θ
(2.144)
Whenever the set of parameters
θ
to be estimated is random, we may
think about any “measuring of closeness” between
θ
and its estimate. A sta-
tistical average of the estimation error is not per se a suitable candidate, since
it is possible, for example, that a zero-mean error has a significant variance.
In other words, the estimator may be unbiased but not efficient. A suitable
option is to work with the statistical average of the square of the error, i.e.,
with the mean-squared error (MSE). Such option originates the method of
the MMSE estimation, which consists in finding the
θ
that minimizes
E
2
E
2
−
θ
(
θ
J
MSE
)
=
=
θ
(2.145)
In (2.145) it should be emphasized that since
θ
is random, the expectation
operator is taken with respect to the joint pdf
p
(
x
,
θ
)
, which means that
−
θ
2
J
MSE
(
θ
)
=
θ
p
(
x
,
θ
)
d
x
d
θ
,
X
d
θ
p
X
(
−
θ
2
=
θ
p
(
θ
|
x
)
x
)
d
x
(2.146)
|
X
