# SVM confidence according to distance from hyperline

For a probabilistic multi-class classifier we can get probabilities of membership of a new point $x$ to each class $y_i$; in case of 3 classes suppose that we get $P(y_a|x) > P(y_b|x) > P(y_c|x)$, thus the most probable class of x is $y_a$. Now suppose that we have a multi-class svm where we can get scores of membership of $x$ to each class (according to distances from hyperlines); in case of 3 classes suppose that we get $Score(y_a|x), Score(y_b|x), Score(y_c|x)$, How is in this case the first, second and third most likely class of $x$ (without converting these scores to probabilities) ? Usually I get positive and negative values like for instance $Score1 = -8622, Score2 = 5233, Score3 = -665$

It's actually possible to get probabilities out of a Support Vector Machine, which might be more useful and interpretable than an arbitrary "score" value. There are a few approaches for doing this: one reasonable place to start is Platt (1999).

Most SVM packages/libraries implement something like this (for example, the -b 1 option causes LibSVM to produce probabilities). If you're going to roll your own, you should be aware that there are some potential numerical issues, summarized in this note by Lin, Lin, and Weng (2007). They also provide some psuedocode, which might be helpful too.

Edit in response to your comment: It's somewhat unclear to me why you'd prefer a score to a probability, especially since you can get the probability with minimal extra effort. All that said, most of the probability calculations seem like they're derived from the distance between the point and the hyperplane. If you look at Section 2 of the Platt paper, he walks through the motivation and says:

The class conditional densities between the margins are apparently exponential. Bayes' rule on two exponentials suggests using a parametric form of a sigmoid: $$P(y=1 | f) = \frac{1}{1+\exp(Af+B)}$$ This sigmoid model is equivalent to assuming that the output of the SVM is proportional to the log-likelihood of a positive training example. [MK: $f$ was defined elsewhere to be the raw SVM output].

The rest of the method section describes how to fit the $A$ and $B$ parameters of that sigmoid. In the introduction (Section 1.0 and 1.1), Platt reviews a few other approaches by Vapnik, Wahba, and Hasti & Tibshirani. These methods also use something like the distance to the hyperplane, manipulated in various ways. These all seem to suggest that the distance to the hyperplane contains some useful information, so I guess you could use the raw distance as some (non-linear) measure of confidence.

• Well, I know there are methods to convert the scores (distances from boundary) to probabilities and that it might be more interpretable, however, as it is clearly stated in the question: I want to use these scores "without converting these scores to probabilities". So my question is: does it make sense to take the absolute values of all these scores and say that the higher is the absolute value of a score $|s_i|$ the most likely $x$ is of class $i$ ? Or should I compare the scores without the absolute values ? or ... ? – shn Apr 4 '13 at 12:08

If the training dataset is reasonably balanced and has standardized features, I will take the SVM scores as the measure of confidence in belonging to the respective classes. The so-called calibration methods that convert the scores to probability-like quantities, such as Platt scaling, usually use monotone functions (like logistic function) to map the scores to probabilities. Hence, if you only want to compare the confidence levels of a learned SVM model in a particular test datapoint belonging to possible classes, you can just compare the score values (not their absolute values) given that the training dataset from which the model is learned is fairly balanced and does not have any unusual quirk.

• I don't understand this answer. – Michael R. Chernick Nov 20 '19 at 3:05
• I edited it to make my point clearer. – Reza Nov 21 '19 at 4:55