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

Now we calculate (a):
p(F|E) = p(E|F)p(F) / (p(E|F)p(F) + p(E|F)p(F)) =
(0.99$10 45 ) / (0.99$10 45 + 0.005$0.99999) = 0.002.
Roughly 0.2% of people who test positive actually have the disease. Getting a
positive should not be an immediate cause for alarm (famous last words).
Now we calculate (b):
p(F|E) = p(E|F)p(F) / (p(E|F)p(F) + p(E|F)p(F))
(0.995$0.99999) / (0.995$0.99999 + 0.01$10 45 ) = 0.9999999.
Thus, 99.99999% of people who test negative really do not have the disease.
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Bayesian Spam Filters used to be the first line of defense for email programs.
Like many good things, the spammers ran right over the process in about two
years. However, it is an interesting example of useful discrete mathematics.
The filtering involves a training period. Email messages need to be marked as
Good or Bad messages, which we will denote as being the G or B sets.
Eventually the filter will mark messages for you, hopefully accurately.
The filter finds all of the words in both sets and keeps a running total of each
word per set. We construct two functions n G (w) and n B (w) that return the
number of messages containing the word w in the G and B sets, respectively.
We use a uniform distribution. The empirical probability that a spam message
contains the word w is p(w) = n B (w) / |B|. The empirical probability that a nonspam
message contains the word w is q(w) = n G (w) / |G|.
We can use p and q to estimate if an incoming message is or is not spam based
on a set of words that we build dynamically over time.
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