Implement Bayesian inference using PHP, Part 1 - Conditional probability and SQL
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P(A | B) can be mapped onto database-query operations. For example, the probability of cancer given a positive test result, P(+cancer | +test), can be obtained by issuing this SQL query then doing some tallies on the result set like this:
SELECT cancer_status FROM Data WHERE test_status='+test'
If I gather information about how several boolean-valued tests co-vary with a boolean-valued diagnosis (like that of cancer or not cancer), then I can perform slightly more complex queries to study how diagnostically useful other factors are in determining whether a patient has cancer, such as in the following:
SELECT cancer_status
FROM Data
WHERE genetic_status='+'
AND age_status='+'
AND biopsy_status='+'
In the case of detecting e-mail spam, I might be interested in computing P(+spam | title_word='viagra' AND title_word='free'), which could be viewed as a directive to issue the following SQL query:
SELECT spam_status FROM Emails WHERE email_title LIKE 'viagra'
AND email_title LIKE 'free'
After enumerating the number of e-mails that are spam and have "viagra" and "free" in the title (like so):
count_emails(spam_status='+spam' AND email_title LIKE 'viagra'
AND email_title LIKE 'free')
and dividing by the overall number of e-mails with the words "viagra" and "free" in the title:
count_emails(email_title LIKE 'viagra' AND email_title LIKE 'free')
I might arrive at the conclusion that the appearence of these words in the title strongly and specifically co-varies with the message being spam (after all, 18/18 = 100 percent) and this rule might be used to automatically filter such messages.
In Bayes spam filtering, you need to initially train the software in which e-mails are spam and which are not. One can imagine storing spam_status information with each e-mail record (for example, email_id, spam_status, email_title, or email_message) and doing the previous queries and counts on this data to decide whether to forward a new e-mail into your inbox.
Next: Frequency versus probability format >>
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