Bayesian message classification

Bayesian spam filtering is the process of using Bayesian statistical methods to classify documents into categories.

Bayesian filtering was proposed by Sahami et al. (1998) and gained attention in 2002 when it was described in a paper by Paul Graham. Since then it has become a popular mechanism to distinguish illegitimate spam email from legitimate email. Many modern mail programs implement Bayesian spam filtering. Server-side email filters, such as SpamAssassin and ASSP, make use of Bayesian spam filtering techniques, and the functionality is sometimes embedded within mail server software itself.

The regular incidence of Spam with selections of lengthy normal text passages from books, etc, is an attempt to corrupt this process by modifying the occurrences of 'non-Spam' words in Spam messages.

Mathematical foundation

Bayesian email filters take advantage of Bayes' theorem. Bayes' theorem, in the context of spam, says that the probability that an email is spam, given that it has certain words in it, is equal to the probability of finding those certain words in spam email, times the probability that any email is spam, divided by the probability of finding those words in any email:

[math]\displaystyle{ \Pr(\mathrm{spam}|\mathrm{words}) = \frac{\Pr(\mathrm{words}|\mathrm{spam})\Pr(\mathrm{spam})}{\Pr(\mathrm{words})}. }[/math]

Process

Particular words have particular probabilities of occurring in spam email and in legitimate email. For instance, most email users will frequently encounter the word Viagra in spam email, but will seldom see it in other email. The filter doesn't know these probabilities in advance, and must first be trained so it can build them up. To train the filter, the user must manually indicate whether a new email is spam or not. For all words in each training email, the filter will adjust the probabilities that each word will appear in spam or legitimate email in its database. For instance, Bayesian spam filters will typically have learned a very high spam probability for the words "Viagra" and "refinance", but a very low spam probability for words seen only in legitimate email, such as the names of friends and family members.

After training, the word probabilities (also known as likelihood functions) are used to compute the probability that an email with a particular set of words in it belongs to either category. Each word in the email contributes to the email's spam probability. This contribution is called the posterior probability and is computed using Bayes' theorem. Then, the email's spam probability is computed over all words in the email, and if the total exceeds a certain threshold (say 95%), the filter will mark the email as spam. Email marked as spam can then be automatically moved to a "Junk" email folder, or even deleted outright.

Advantages

The advantage of Bayesian spam filtering is that it can be trained on a per-user basis.

The spam that a user receives is often related to the online user's activities. For example, a user may have been subscribed to an online newsletter that the user considers to be spam. This online newsletter is likely to contain words that are common to all newsletters, such as the name of the newsletter and its originating email address. A Bayesian spam filter will eventually assign a higher probability based on the user's specific patterns.

The legitimate e-mails a user receives will tend to be different. For example, in a corporate environment, the company name and the names of clients or customers will be mentioned often. The filter will assign a lower spam probability to emails containing those names.

The word probabilities are unique to each user and can evolve over time with corrective training whenever the filter incorrectly classifies an email. As a result, Bayesian spam filtering accuracy after training is often superior to pre-defined rules.

It can perform particularly well in avoiding false positives, where legitimate email is incorrectly classified as spam. For example, if the email contains the word "Nigeria", which frequently appeared in a long spam campaign, a pre-defined rules filter might reject it outright. A Bayesian filter would mark the word "Nigeria" as a probable spam word, but would take into account other important words that usually indicate legitimate e-mail. For example, the name of a spouse may strongly indicate the e-mail is not spam, which could overcome the use of the word "Nigeria."

Some spam filters combine the results of both Bayesian spam filtering and pre-defined rules resulting in even higher filtering accuracy. Recent spammer tactics include insertion of random innocuous words that are not normally associated with spam, thereby decreasing the email's spam score, making it more likely to slip past a Bayesian spam filter.

General applications of Bayesian filtering

While Bayesian filtering is used widely to identify spam email, the technique can classify (or "cluster") almost any sort of data. It has uses in science, medicine, and engineering. One example is a general purpose classification program called AutoClass which was originally used to classify stars according to spectral characteristics that were otherwise too subtle to notice. There is recent speculation that even the brain uses Bayesian methods to classify sensory stimuli and decide on behavioural responses (Trends in Neuroscience, 27(12):712-9, 2004) (pdf).

See also

• Bayesian inference
• Bayes' theorem
• Naive Bayes classifier
• Recursive Bayesian estimation
• Stopping e-mail abuse

External links

• Guide to Bayesian spam filters: part 1, part 2.
• SpamBayes - Open source Bayesian spam filter.
• Detailed explanation of Paul Graham's formulas, which lack mathematical rigorousness

References

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