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Old 01-30-2013, 01:09 PM
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yaser yaser is offline
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Location: Pasadena, California, USA
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Default Re: VC dimension independent of probability distribution

Originally Posted by Anne Paulson View Post
But now I'm confused. Are we assuming that we use the same probability distribution when computing the error rate on the whole input space? That is, we check the error on every single point, but the ones that were more likely to be in the training set get weighted more, so it's an expectation over all possible training sets picked using the probability distribution, rather than just an error rate over the entire input space with uniform distribution?

Otherwise it doesn't make sense to me. Seems like we could rig the training set to make our cockamamie hypothesis look good.
The source of randomization in the VC inequality is the choice of the training set {\cal D}. The assumption is that this set is generated according to some probability distribution P on X, independently from one data point in {\cal D} to the other. The same probability distribution is used to compute E_{\rm out} by averaging over the whole input space {\cal X}.

Now, the statement that the VC inequality is independent of the probability distribution means that it holds for any probability distribution. Any training set you pick, according to any probability distribution (or even a rigged training set for that matter) will have at most as many dichotomies as the value of the growth function, since the growth function is defined as a maximum. Since that value is what is needed for the proof of the VC inequality, the inequality will always hold (more loosely for certain probability distributions, but will nonetheless hold).

Having said that, we are still not allowed to "rig" the choice of the training set {\cal D}, not because the number of dichotomies will be a problem -- it won't be, but because the basic premise of Hoeffding, on which the VC inequality is built, is that the points in {\cal D} are picked independently according to some probability distribution. You can rig the probability distribution if you want, but you still have to pick your data points independently from it, and use the same probability distribution to compute E_{\rm out}.
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