Learning Data Mining with Python

Now, we can compute the probability of the data point belonging to this class.
An important point to note is that we haven't computed P(D), so this isn't a real
probability. However, it is good enough to compare against the same value for
the probability of the class 1. Let's take a look at the calculation:
P(C=0|D) = P(C=0) P(D|C=0)
= 0.75 * 0.0756
= 0.0567
Now, we compute the same values for the class 1:
P(C=1) = 0.25
P(D) isn't needed for naive Bayes. Let's take a look at the calculation:
P(D|C=1) = P(D1|C=1) x P(D2|C=1) x P(D3|C=1) x P(D4|C=1)
= 0.7 x 0.7 x 0.6 x 0.9
= 0.2646
P(C=1|D) = P(C=1)P(D|C=1)
= 0.25 * 0.2646
= 0.06615
Chapter 6
Normally, P(C=0|D) + P(C=1|D) should equal to 1. After all, those
are the only two possible options! However, the probabilities are
not 1 due to the fact we haven't included the computation of P(D)
in our equations here.
The data point should be classified as belonging to the class 1. You may have
guessed this while going through the equations anyway; however, you may have
been a bit surprised that the final decision was so close. After all, the probabilities
in computing P(D|C) were much, much higher for the class 1. This is because we
introduced a prior belief that most samples generally belong to the class 0.
If the classes had been equal sizes, the resulting probabilities would be much
different. Try it yourself by changing both P(C=0) and P(C=1) to 0.5 for equal class
sizes and computing the result again.
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