A Statistical Approach to the Spam Problem
Most people are spending significant time daily on the task of distinguishing spam from useful e-mail. I don't think I'm alone in feeling that we have better things to do. Spam-filtering software can help.
This article discusses one of many possible mathematical foundations for a key aspect of spam filtering—generating an indicator of “spamminess” from a collection of tokens representing the content of an e-mail.
The approach described here truly has been a distributed effort in the best open-source tradition. Paul Graham, an author of books on Lisp, suggested an approach to filtering spam in his on-line article, “A Plan for Spam”. I took his approach for generating probabilities associated with words, altered it slightly and proposed a Bayesian calculation for dealing with words that hadn't appeared very often. Then I suggested an approach based on the chi-square distribution for combining the individual word probabilities into a combined probability (actually a pair of probabilities—see below) representing an e-mail. Finally, Tim Peters of the Spambayes Project proposed a way of generating a particularly useful spamminess indicator based on the combined probabilities. All along the way the work was guided by ongoing testing of embodiments written in Python by Tim Peters for Spambayes and in C by Greg Louis of the Bogofilter Project. The testing was done by a number of people involved with those projects.
We will assume the existence of a body of e-mails (the corpus) for training, together with software capable of parsing each e-mail into its constituent words. We will further assume that each training e-mail has been classified manually as either “ham” (the e-mail you want to read) or “spam” (the e-mail you don't). We will use this data and software to train our system by generating a probability for each word that represents its spamminess.
For each word that appears in the corpus, we calculate:
b(w) = (the number of spam e-mails containing the word w) / (the total number of spam e-mails).
g(w) = (the number of ham e-mails containing the word w) / (the total number of ham e-mails).
p(w) = b(w) / (b(w) + g(w))
p(w) can be roughly interpreted as the probability that a randomly chosen e-mail containing word w will be a spam. Spam-filtering programs can compute p(w) for every word in an e-mail and use that information as the basis for further calculations to determine whether the e-mail is ham or spam.
However, there is one notable wrinkle. In the real world, a particular person's inbox may contain 10% spam or 90% spam. Obviously, this will have an impact on the probability that, for that individual, an e-mail containing a given word will be spam or ham.
That impact is ignored in the calculation above. Rather, that calculation, in effect, approximates the probabilitily that a randomly chosen e-mail containing w would be spam in a world where half the e-mails were spams and half were hams. The merit of that approach is based on the assumption that we don't want it to be harder or easier for an e-mail to be classified as spam just because of the relative proportion of spams and hams we happen to receive. Rather, we want e-mails to be judged purely based on their own characteristics. In practice, this assumption has worked out quite well.
There is a problem with probabilities calculated as above when words are very rare. For instance, if a word appears in exactly one e-mail, and it is a spam, the calculated p(w) is 1.0. But clearly it is not absolutely certain that all future e-mail containing that word will be spam; in fact, we simply don't have enough data to know the real probability.
Bayesian statistics gives us a powerful technique for dealing with such cases. This branch of statistics is based on the idea that we are interested in a person's degree of belief in a particular event; that is the Bayesian probability of the event.
When exactly one e-mail contains a particular word and that e-mail is spam, our degree of belief that the next time we see that word it will be in a spam is not 100%. That's because we also have our own background information that guides us. We know from experience that virtually any word can appear in either a spam or non-spam context, and that one or a handful of data points is not enough to be completely certain we know the real probability. The Bayesian approach lets us combine our general background information with the data we have collected for a word in such a way that both aspects are given their proper importance. In this way, we determine an appropriate degree of belief about whether, when we see the word again, it will be in a spam.
We calculate this degree of belief, f(w), as follows:
s is the strength we want to give to our background information.
x is our assumed probability, based on our general backround information, that a word we don't have any other experience of will first appear in a spam.
n is the number of e-mails we have received that contain word w.
This gives us the convenient use of x to represent our assumed probability from background information and s as the strength we will give that assumption. In practice, the values for s and x are found through testing to optimize performance. Reasonable starting points are 1 and .5 for s and x, respectively.
We will use f(w) rather than p(w) in our calculations so that we are working with reasonable probabilities rather than the unrealistically extreme values that can often occur when we don't have enough data. This formula gives us a simple way of handling the case of no data at all; in that case, f(w) is exactly our assumed probability based on background information.
Those who are interested in the derivation of the above formula, read on; others may want to skip down to the next section.
If you are already familiar with the principles of Bayesian statistics, you will probably have no trouble understanding the derivation. Otherwise, I suggest that you read sections 1 and 2 of David Heckerman's “A Tutorial on Learning with Bayesian Networks” (see Resources) before continuing.
The formula above is based on assuming that spam/ham classification for e-mails containing word w is a binomial random variable with a beta distribution prior. We calculate the posterior expectation after incorporating the observed data. We are going to do a test that is the equivalent of the “experiment” of tossing a coin multiple times to see whether it appears to be biased. There are n trials. If we were tossing a coin, each coin toss would be a trial, and we'd be counting the number of heads. But in our case, the trial is looking at the next e-mail in the training corpus that contains the word “porn” and seeing whether the e-mail it contains is a spam. If it is, the experiment is considered to have been successful. Clearly, it's a binomial experiment: there are two values, yes or no. And they are independent: the fact that one e-mail contains “porn” isn't correlated to the question of whether the next one will. Now, it happens that if you have a binomial experiment and assume a beta distribution for the prior, you can express the expectation that the nth + 1 trial will be successful as shown in Equation 2.
q is the number of successes.
n is the number of trials.
u and v are the parameters of the beta distribution.
We want to show that Equation 1 is equivalent to Equation 2. First perform the following substitutions:
s = u + v s * x = u
The next step is to replace q with n * p(w). We have already discussed the fact that p(w) is an approximation of the probability that a randomly chosen e-mail containing w is spam in an imaginary world where there are the same number of spams as hams. So n * p(w) approximates the count of spams containing w in that world. This approximates the count of successes in our experiment and is therefore the equivalent of q. This completes our demonstration of the equivalence of Equations 1 and 2.
In testing, replacing p(w) with f(w) in all calculations where p(w) would otherwise be used, has uniformly resulted in more reliable spam/ham classification.