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

Let E be the event that an incoming message contains the word w. Let S be the
event that an incoming message is spam and contains the word w. Bayes
theorem tells us that the probability that an incoming message containing the
word w is spam is
p(S|E) = p(E|S)p(S) / (p(E|S)p(S) + p(E|S)p(S)).
If we assume that p(S) = p(S) = 0.5, i.e., that any incoming message is equally
likely to be spam or not, then we get the simplified formula
p(S|E) = p(E|S) / (p(E|S) + p(E|S)).
We estimate p(E|S) = p(w) and p(E|S) = q(w). So, we estimate p(S|E) by
r(w) = p(w) / (p(w) + q(w)).
If r(w) is greater than some preset threshold, then we classify the incoming
message as spam. We can consider a threshold of 0.9 to begin with.
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Example: Let w = Rolex. Suppose it occurs in 250 / 2000 spam messages and in
5 / 1000 good messages. We will estimate the probability that an incoming
message with Rolex in it is spam assuming that it is equally likely that the
incoming message is spam or not. We know that p(Rolex) = 250 / 2000 = 0.125
and q(Rolex) = 5 / 1000 = 0.005. So,
r(Rolex) = 0.125 / (0.125 + 0.005) = 0.962 > 0.9.
Hence, we would reject the message as spam. (Note that some of us would reject
all messages with the word Rolex in it as spam, but that is another case entirely.)
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