Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

Using just one word to determine if a message is spam or not leads to excessive
numbers of false positives and negatives. We actually have to use the
generalized Bayes theorem with a large set of words.
k
i=1
E i
p(S | ! ) =
k
i=1
!
k
i=1
p(E i
|S)
k
i=1
! p(E i
|S) + ! p(E i
|S)
,
which we estimate assuming equal probability that an incoming message is
spam or not by
r(w 1
,w 1
,...,w 1
) =
!
k
i=1
p(w i
)
.
k
k
! p(w i
) + ! q(w i
)
i=1
i=1
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Example: The word w 1 = stock appears in 400 / 2000 spam messages and in just
60 / 1000 good messages. The word w 2 = undervalued appears in 200 / 2000
spam messages and in just 25 / 1000 good messages. Estimate the likelihood that
an incoming message with both words in it is spam. We know p(stock) = 0.2 and
q(stock) = 0.06. Similarly, p(undervalued) = 0.1 and q(undervalued) = .025. So,
r(stock,undervalued) =
p(stock)p(undervalued)
p(stock)p(undervalued)+q(stock)q(undervalued)
=
0.2!0.1
0.2!0.1+0.06!0.025
= 0.930 > 0.9
Note: Looking for particular pairs or triplets of words and treating each as a
single entity is another method for filtering. For example, enhance performance
probably indicates spam to almost anyone, but high performance computing
probably does not indicate spam to someone in computational sciences (but
probably will for someone working in, say, Maytag repair).
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