- Transforming Classifier Scores into Accurate Multiclass Probability Estimates, Zadrozny/Elkan, SIGKDD 2002
- Obtaining calibrated probability estimates from decision trees and naive Bayesian classifiers, Zadrozny/Elkan
- Predicting Good Probabilities With Supervised Learning, Niculescu-Mizil/Caruana.
The overall idea is that the scores output by a classifier need to be calibrated so that they can be understood. And, specifically, if you want to understand them as a probability that a document falls into a particular category then calibration gives you a way to estimate the probability from the scores.
The simplest technique is bucketing or binning. Suppose that the classifier outputs a score s(x) in the range [0,1) for an input document x. Once a classifier is trained it's possible to calibrating it by classifing known documents, recording the output scores and then for each of a set of ranges (e.g. divide [0,1) into 10 ranges [0.0,0.1), [0.1,0.2) and so on) count the number of documents in each class, and get a fraction the represents the probability within that range. Then when you classify an unknown document you look at the range in which its score falls to get probability estimates.
For spam filtering each range would consist of the number of emails that are ham in the range, and the number that are spam. The ham probability then just comes out to # of ham emails in the range / total number of emails in the range.
I decided to run a little test of this and took the complete TREC 2005 Public Spam Corpus and trained a Gary Robinson style spam filter on it. Then I classified the entire corpus to get calibration data. Instead of looking at the final scores, I looked at the pair of scores (one for ham, one for spam) that the classifier generates and used those to generate bins. The following chart shows (crudely) the percentage of spams and hams in each bin:
The x-axis is the ham score in the range [0.0,0.9) with each square representing a single 0.1-width bin. The left-most square means that the classifier had a very low score in terms of hamminess, right-most square means that the classifier had a very high score.
The y-axis is the spam score in the [0.0,0.9), so that the bottom means a low spamminess score and the top a high one. Each square is colored red and green. Red is the percentage of messages in that bin that are spam, and green the percentage of messages in that bin that are ham.
So as you'd expect the top left corner (low ham score, high spam score) is generally colored red indicating that these are spam messages, and the bottom right corner (low spam score, high ham score) is colored green (lots of ham). Some squares are black because there's too little data (I arbitrarily said that if there were less than 5 messages in the bin it was colored black).
But there's a really interesting square in the top right: it's almost all red (in fact, it's 93% red or spam). So we can have confidence that a message there (which corresponds to a final 'score' of 0.5) is in fact spam.
I'm sure that there are other interesting insights to be gained from calibration and I'd like to spend the time to evaluate the effectiveness of using the probabilities generated in this way (instead of the rather arbitrary selection of a score 'cutoff' point) as the way to determine whether a message is spam or ham (or undetermined). For example, the square at (0.9,0.2) (which corresponds to a 'score' of 0.18 is 57% ham, 43% spam so looks like a good candidate for undetermined; it looks like a much better candidate than the typical "area around 0.5" (which corresponds to (0.9,0.9) and is 93% spam).