## 5. Bayesian word distribution filters

Paul Graham wrote a provocative essay in August 2002. In "A Plan for Spam" (see Resources later in this article), Graham suggested building Bayesian probability models of spam and non-spam words. Graham's essay, or any general text on statistics and probability, can provide more mathematical background than I will here.

The general idea is that some words occur more frequently in known spam, and other words occur more frequently in legitimate messages. Using well-known mathematics, it is possible to generate a "spam-indicative probability" for each word. Another simple mathematical formula can be used to determine the overall "spam probability" of a novel message based on the collection of words it contains.

Graham's idea has several noteworthy benefits:

- It can generate a filter automatically from corpora of categorized messages rather than requiring human effort in rule development.
- It can be customized to individual users' characteristic spam and legitimate messages.
- It can be implemented in a very small number of lines of code.
- It works surprisingly well.

At first blush, it would be reasonable to suppose that a set of hand-tuned and laboriously developed rules like those in SpamAssassin would predict spam more accurately than a scattershot automated approach. It turns out that this supposition is dead wrong. A statistical model basically just works better than a rule-based approach. As a side benefit, a Graham-style Bayesian filter is also simpler and faster than SpamAssassin.

Within days -- perhaps hours -- of Graham's article being published, many people simultaneously started working on implementing the system. For purposes of my testing, I used a Python implementation created by a correspondent of mine named John Barham. I thank him for providing his implementation. However, the mathematics are simple enough that every other implementation is largely equivalent.

There are some issues of data structures and storage techniques that will effect operating speed of different tools. But the actual predictive accuracy depends on very few factors -- the main significant factor is probably the word-lexing technique used, and this matters mostly for eliminating spurious random strings. Barham's implementation simply looks for relatively short, disjoint sequences of characters in a small set (alphanumeric plus a few others).

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