I have learned that, when dealing with data using model-based approach, the first step is modeling data procedure as a statistical model. Then the next step is developing efficient/fast inference/learning algorithm based on this statistical model. So I want to ask which statistical model is behind the support vector machine (SVM) algorithm?


2 Answers 2


You can often write a model that corresponds to a loss function (here I'm going to talk about SVM regression rather than SVM-classification; it's particularly simple)

For example, in a linear model, if your loss function is $\sum_i g(\varepsilon_i) = \sum_i g(y_i-x_i'\beta)$ then minimizing that will correspond to maximum likelihood for $f\propto \exp(-a\,g(\varepsilon))$ $= \exp(-a\,g(y-x'\beta))$. (Here I have a linear kernel)

If I recall correctly SVM-regression has a loss function like this:

plot of epsilon-insensitive loss

That corresponds to a density that is uniform in the middle with exponential tails (as we see by exponentiating its negative, or some multiple of its negative).

plot of corresponding density

There's a 3 parameter family of these: corner-location (relative insensitivity threshold) plus location and scale.

It's an interesting density; if I recall rightly from looking at that particular distribution a few decades ago, a good estimator for location for it is the average of two symmetrically-placed quantiles corresponding to where the corners are (e.g. midhinge would give a good approximation to MLE for one particular choice of the constant in the SVM loss); a similar estimator for the scale parameter would be based on their difference, while the third parameter corresponds basically to working out which percentile the corners are at (this might be chosen rather than estimated as it often is for SVM).

So at least for SVM regression it seems pretty straightforward, at least if we're choosing to get our estimators by maximum likelihood.

(In case you're about to ask ... I have no reference for this particular connection to SVM: I just worked that out now. It's so simple, however, that dozens of people will have worked it out before me so no doubt there are references for it -- I've just never seen any.)

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    $\begingroup$ (I answered this earlier elsewhere but I deleted that and moved it here when I saw you also asked here; the ability to write mathematics and include pictures is much better here -- and the search function is better too, so it's easier to find in a few months) $\endgroup$
    – Glen_b
    Commented Nov 14, 2017 at 7:44
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    $\begingroup$ +1, plus the vanilla SVM also has a Gaussian prior on its parameters through the $\ell_2$-norm. $\endgroup$
    – Firebug
    Commented Nov 14, 2017 at 10:15
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    $\begingroup$ If the OP is asking about SVM, s/he is probably interested in classification (which is the most common application of SVMs). In that case the loss is hinge loss which is a bit different (you don't have the increasing part). Concerning the model, I heard academics saying at conference that SVMs were introduced to perform classification without having to use a probabilistic framework. Probably that's why you can't find references. On the other hand, you can and you do recast hinge loss minimization as empirical risk minimization - which means... $\endgroup$
    – DeltaIV
    Commented Nov 14, 2017 at 10:32
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    $\begingroup$ Just because you don't have to have a probabilistic framework... doesn't mean what you're doing doesn't correspond to one. One can do least squares without assuming normality, but it's useful to understand that's what it's doing well at ... and when you're nowhere near it that's it may be doing much less well. $\endgroup$
    – Glen_b
    Commented Nov 14, 2017 at 10:34
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    $\begingroup$ Maybe icml-2011.org/papers/386_icmlpaper.pdf is a reference for this?(I've only skimmed it) $\endgroup$ Commented Nov 15, 2017 at 6:52

I think someone already answered your literal question, but let me clear up a potential confusion.

Your question is somewhat similar to the following:

I have this function $f(x) = \ldots$ and I'm wondering what differential equation it is a solution to?

In other words, it certainly has a valid answer (perhaps even a unique one if you impose regularity constraints), but it's a rather strange question to ask, since it was not a differential equation that gave rise to that function in the first place.
(On the other hand, given the differential equation, it is natural to ask for its solution, since that's usually why you write the equation!)

Here's why: I think you're thinking of probabilistic/statistical models—specifically, generative and discriminative models, based on estimating joint and conditional probabilities from data.

The SVM is neither. It's an entirely different kind of model—one that bypasses those and attempts to directly model the final decision boundary, the probabilities be damned.

Since it's about finding the shape of the decision boundary, the intuition behind it is geometric (or perhaps we should say optimization-based) rather than probabilistic or statistical.

Given that probabilities aren't really considered anywhere along the way, then, it's rather unusual to ask what a corresponding probabilistic model could be, and especially since the entire goal was to avoid having to worry about probabilities. Hence why you don't see people talking about them.

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    $\begingroup$ I think you are discounting the value of statistical models underlying your procedure. The reason it is useful is that it tells you what assumptions are behind a method. If you know these, you are able to understand which situations it will struggle and when it will thrive. You are also able to generalise and extend svm in a principled manner if you have the underlying model. $\endgroup$ Commented Nov 14, 2017 at 10:05
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    $\begingroup$ @probabilityislogic: "I think you are discounting the value of statistical models underlying your procedure." ...I think we're speaking past each other. What I am trying to say is that there isn't a statistical model behind the procedure. I am not saying that it's not possible to come up with one that fits it a posteriori, but I'm trying to explain that it was not "behind" it in any way, but rather "fit" to it after the fact. I am also not saying that doing such a thing is useless; I agree with you that it could end up with tremendous value. Please bear these distinctions in mind. $\endgroup$
    – user541686
    Commented Nov 14, 2017 at 10:35
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    $\begingroup$ @Mehrdad: I am not saying that it's not possible to come up with one that fits it a posteriori, The order in which the pieces of what we call the svm 'machine' were assembled (what problem the humans who designed it were originally trying to solve) is interesting from an history of science point of view. But for all we know there might be an yet unknown manuscript in some library containing a description of the svm engine from 200 years ago which attacks the problem from the angle Glen_b explored. Maybe the notions of a posteriori and after the fact are less dependable in science. $\endgroup$
    – user603
    Commented Nov 14, 2017 at 12:37
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    $\begingroup$ @user603: It's not just the history that's the problem here. The historical aspect is only half of it. The other half is how it's normally actually derived in reality. It starts as a geometry problem and ends with an optimization problem. Nobody starts with the probabilistic model in the derivation, meaning the probabilistic model was in no sense "behind" the result. It's like claiming Lagrangian mechanics is "behind" F = ma. Maybe it can lead to it, and yes it is useful, but no, it is not and never was the basis of it. In fact the entire goal was to avoid probability. $\endgroup$
    – user541686
    Commented Nov 14, 2017 at 19:43

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