Correlated probabilities? Try a logarithm.

Suppose that we have a patient and are trying to determine whether his leg is broken. A general practitioner investigates the patient and a radiologist looks at a scan of his leg. Both doctors give their conclusion in the form of a probability that the leg is broken, \(P_{GP}\) and \(P_{R}\). How do we combine these probabilities into one? If these were independent then we could just multiply \(P_{GP}\cdot P_{R}\), but it’s likely that both the GP and the radiologist will have the same opinion, so they are not independent. Furthermore, maybe the opinion of the radiologist is more accurate than the opinion of the general practitioner, because the radiologist looks at a scan of the leg.

We could forget that these are probabilities altogether, and focus on the decision of whether to operate the patient or not. We assign a combined score \(f(P_{GP},P_{R})\) in some way, and then look empirically for a decision boundary \(f(P_{GP},P_{R})<\alpha\) that gives us a trade-off between the false positive and false negative rate. The question remains which \(f\) we should use.

A linear model is usually the first you’d try, \(f(P_{GP},P_{R})=aP_{GP}+bP_{R}\), but I claim that \(f(P_{GP},P_{R})=P_{GP}^{a}\cdot P_{R}^{b}\) is more natural. If the probabilities were independent then \(a=1,b=1\) would give \(P_{GP}\cdot P_{R}\). Choosing different \(a,b\) weighs the opinions, e.g. \(a=1/3\), \(b=2/3\).

This is equivalent to training a linear model on the log probabilities \(\log P_{GP}\) and \(\log P_{R}\), because \(\log(P_{GP}^{a}\cdot P_{R}^{b})=a\log P_{GP}+b\log P_{R}\). The log probability is natural from the point of view of information theory: log probability is measured in bits. Probabilities get multiplied, bits get added.

If you’re training a classifier on probabilities, try a logarithm.