I have training data that falls into two classes, let's say Yes and No. The data represents three tasks, easy, medium and difficult. A person performs these tasks and is classified into one of the two classes as a result. Each task is classified independently and then the results are combined. I am using 3 independently trained SVM classifiers and then voting on the final result.
I am looking to provide a measure of confidence or probability associated with each classification. LIBSVM can provide a probability estimate along with the classification for each task (easy, medium and difficult, say Pe, Pm and Pd) but I am unsure of how best to combine these into an overall estimate for the final classification of the person (let's call it Pp).
My attempts so far have been along the lines of a simple average:
Pp = (Pe + Pm + Pd) / 3
An Inverse-variance weighted average (since each task is repeated a few times and sample variance (VARe, VARm and VARd) can be calculated - in which case Pe would be a simple average of all the easy samples):
Pp = (Pe/VARe + Pm/VARm + Pd/VARd) / (( 1/VARe ) + ( 1/VARm ) + ( 1/VARd ))
Or a multiplication (under the assumption that these events are independent, which I am unsure of since the underlying tasks are related):
Pp = Pe * Pm * Pd
The multiplication would provide a very low number, so it's unclear how to interpret that as an overall probability when the results of the voting are very clear.
Would any of these three options be the best or is there some other method / detail I'm overlooking?
Based on your comment, I will make the following suggestion. If you need to do this as an SVM (and because, as you say, you get better performance when you do it this way), take the output from your intermediate classifiers and feed them as features to your final classifier. Even better, switch to a multi-layer Neural Net where your inputs represent inputs to the intermediates, the (first) hidden layer represents outputs to the intermediate problem, and subsequent layer(s) represent the final decision you want. This way you get the benefit of an intermediate layer, but its output is optimised to help with the final prediction rather than for accuracy in its own right (which I assume you don't really care about).
The correct generative model for these tests likely looks something like the following:
This is simplified, but I think it should illustrate tht you have a latent variable i on which these tests depend (and there's also structure between them, since presumably p_easy(pass|i) > p_medium(pass|i) > p_hard(pass|i); you could potentially model this as a logistic regression with a continuous 'hardness' feature). I suspect what you're asking about is a way to do inference on some thresholding function of i, but you want to do it in a classification way rather than as a probabilistic model. That's fine, but without explicitly encoding the latent variable and the structure between the tests it's going to be hard (and no average of the probabilities will account for the missing structure).
I hope that helps---if I've made assumptions that aren't justified, please feel free to correct.