Hello. I have four different images (X1, X2, X3 and X4) which I classify with four different discriminate probabilistic models (discriminative classifiers) to obtain posterior probabilities of a pixel belonging to four classes i.e. in each case I obtain a vector four 4 indication membership values to the four classes. Assuming C1, C2, C3 and C4 to be class labels then the estimated posterior probabilities in each case as (P(C1|X1), P(C2| X2), P(C2|X3) and P(C4|X4). I wish to combine the probabilities from the 4 classifiers. So assume P(C1), P(C2), P(C3) and P(C4) to represent the posterior probability values (and class conditional independence) I decided to determine their joint probability P(C1,C2,C3,C4) which i solve using chain rule as: P(C1,C2,C3,C4) = P(C1)*P(C1,C2)/P(C1)*P(C1,C2, C3)/P(C1, C2)*P(C1,C2, C3, C4)/P(C1, C2, C3). Is this justifiable theoretically? I have gone through this article (http://www.mpia.de/Gaia/publications/probcomb_TN.pdf) but still did not get my case there clearly. Kindly advice.
Thanks
Hi Benson,
Thanks for yours. There are various ways of doing this:
1. You could make a weighted combination of the four classifiers, simplest is to give them all a weight 1/4, but you ay have reasons to deviate from this. Youcould, for example, spoecify your prior believe in the quality of each classifier and combine that with the prior on the data.
2. You could make a joint, four-dimensional, distribution and maximize that.
Would any of these work?
Alfred
PS, your questions may become better understandable if you start with two classifiers, and then later extend this to four.
Hallo Prof. Stein,
Its great to get an answer from you. I had actually tested the second option that you gave above. That is, I took a joint distribution, P(C1,C2,C3,C4), of the posteriors from the four classifiers using chain rule as illustrated below, and found the maximum probability for a class. The performance is good so far.
Joint distribution by chain rule
P(C1,C2,C3,C4) = P(C1)*P(C1,C2)/P(C1)*P(C1,C2, C3)/P(C1, C2)*P(C1,C2, C3, C4)/P(C1, C2, C3).
However,, I also think some form of weighting is necessary as the final accuracy of the joint distribution gives an average accuracy of the accuracies of the single classifiers.
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