Digital Signal Processing Reference
In-Depth Information
TABLE 2.1
Result of a Hypothetical Pool about the Most Noticeable Flavor Found in
the Wine
Blackcurrant (%)
Plum (%)
Raspberry (%)
Other (%)
Cabernet-Sauvignon
41
45
2
12
Tannat
4
0
79
17
Malbec
31
55
0
14
Merlot
40
30
20
10
grapes traditionally found in this region: Cabernet-Sauvignon, Tannat, Malbec and
Merlot. We can state that the
taste
stands for the observed data while the
grape
is
the parameter to be estimated.
As a support to his difficult task, the taster relies on the information contained
in Table 2.1, which shows the results of a poll in which renowned wine profes-
sionals were asked what is the most noticeable flavor in the wines made by this
producer.
Let us say that the mentioned taster found a very distinguished touch of plum
in the wine being tested. According to Table 2.1, we have that
⎨
⎩
0.45, for variety = Cabernet-Sauvignon
0, for variety = Tannat
0.55, for variety = Malbec
0.30, for variety = Merlot
P
(
taste = plum
|
variety
)
=
Therefore, if the decision is made using the ML criterion, according to (2.139),
the “estimated grape” is the variety that
variety
=
variety
P
max
(
taste = plum
|
variety
)
(2.142)
That is, based on the poll provided by specialists, the taster concludes that, since
the taste of plum is predominant, the wine is more likely to be a Malbec.
Now, suppose that, in addition to Table 2.1, the taster is also informed that
approximately 40% of the total grape production of this wine maker consists of the
cabernet-sauvignon variety, and each one of the other three grapes correspond to
20% of the total production. Since we have prior information, it is reasonable to
employ the MAP criterion in this case. In other words, we take into account not
only the fact that the flavor of plum is claimed by the specialists to be more likely
found if the wine is Malbec, but, since it can also be found in other varieties, we
take in account the probability of occurrence of each variety. Hence, the estimate
is given by
P
(
variety
|
taste = plum
)
∝
P
(
taste = plum
|
variety
)
P
(
variety
)
(2.143)
