# Combining posterior probabilities from multiple classifiers

I am new to machine learning and can't get my head around this problem. I have two patient datasets, the first ($D_1$) contains $Y,Z,X$ that convey blood-sample information and the second ($D_2$) contains $W,T,X$ that convey x-ray information. In both datasets, $X$ is the common diagnostic output.

Since there are two distinct datasets, I modelled a solution with two Naive Bayes models as below, where $X$ is the common dependent variable. ( I am aware I can use other techniques than Naive Bayes but that is not the aim of my question.)

• $P(X|Y,Z,D_1) \propto P(Y|X,D_1) \cdot P(Z|X,D_1) \cdot P(X,D_1)$
• $P(X|W,T,D_2) \propto P(W|X,D_2) \cdot P(T|X,D_2) \cdot P(X,D_2)$

I want to combine the posterior probabilities of X in these models: $P(X|Y,Z)$ and $P(X|W,T)$ to determine an overall outcome probability $P(X|Y,Z,W,T)$ (diagnostic outcome) of a new patient.

How can I combine these probabilities?

Can it be done as below?

$P(X|Y,Z,W,T) \propto \frac{P(X|Y,Z) * P(X|W,T)}{P(X,D_{combined})}$

If so, what is the relationship between the priors $P(X,D_1)$, $P(X,D_2)$ and $P(X,D_{combined})$?

• Check this out - mpia-hd.mpg.de/Gaia/publications/probcomb_TN.pdf – TenaliRaman Nov 4 '13 at 9:55
• Thanks, I am aware of this. I do not understand how the priors from the different datasets $D_A$ and $D_B$ (or in my case $D_1$ and $D_2$) relate to each other though. – user31230 Nov 4 '13 at 17:21
• In fact, they are not related. Or rather, there seems to be no way to know how just like that. If you have two different populations, then you have to guess what the actual prior would be. Even if the output X is the same, the information you have is about the method of determination in light of the given sample. You have a lot of space to argue for any choice you might make on the relations of the priors. But this depends entirely on the datasets and what you know about them. – cherub Sep 13 '19 at 14:47