If you don't get any information after the fact on whether O was Q or not, there's not one right way to do it. JRMayne's recommendation of averaging the expert judgments works, as does DanielLC's recommendation of assuming that the experts are entirely uncorrelated. The trouble with assuming they're uncorrelated is that it can give you pretty extreme probability estimates- but if you're just making decisions based on some middling threshold ("call it a Q if P(Q)>.5") then you don't have to worry about extreme probability estimates! If you make decisions based on an extreme threshold ("call it a Q if P(Q)>.99"), then you have to worry. One of the things that might be helpful is plotting what these formula will result in A,B space, and seeing if that graph looks like what you / experts in this domain would expect.
If you do get information after the fact, you'll want to use what's called a Bayesian Judge. Basically, it learns P(Q(O)|A,B,P(Q)) through Bayesian updates; you're building an expert that says "if I consider all of the n times A said a and B said b, nP times it turned out to be Q, so P(Q)=P."
The other neat thing about Bayesian judges is that they fix calibration problems with experts- it will quickly learn that when they say .9, they actually mean .7.
The trouble with the Bayesian judge is that it will starve if you can't feed it data on whether or not O was Q. I won't type up the necessary math unless this fits your situation, but if it does I'd be happy to.
The trouble with assuming they're uncorrelated is that it can give you pretty extreme probability estimates
No. The trouble with assuming they're uncorrelated is that they probably aren't. If they were, the extreme probability estimates would be warranted.
I suppose more accurately, the problem is that if there is a significant correlation, assuming they're uncorrelated will give a, equally significant error, and they're usually significantly correlated.
Suppose you have a property Q which certain objects may or may not have. You've seen many of these objects; you know the prior probability P(Q) that an object has this property.
You have 2 independent measurements of object O, which each assign a probability that Q(O) (O has property Q). Call these two independent probabilities A and B.
What is P(Q(O) | A, B, P(Q))?
To put it another way, expert A has opinion O(A) = A, which asserts P(Q(O)) = A = .7, and expert B says P(Q(O)) = B = .8, and the prior P(Q) = .4, so what is P(Q(O))? The correlation between the opinions of the experts is unknown, but probably small. (They aren't human experts.) I face this problem all the time at work.
You can see that the problem isn't solvable without the prior P(Q), because if the prior P(Q) = .9, then two experts assigning P(Q(O)) < .9 should result in a probability lower than the lowest opinion of those experts. But if P(Q) = .1, then the same estimates by the two experts should result in a probability higher than either of their estimates. But is it solvable or at least well-defined even with the prior?
The experts both know the prior, so if you just had expert A saying P(Q(O)) = .7, the answer must be .7 . Expert B's opinion B must revise the probability upwards if B > P(Q), and downwards if B < P(Q).
When expert A says O(A) = A, she probably means, "If I consider all the n objects I've seen that looked like this one, nA of them had property Q."
One approach is to add up the bits of information each expert gives, with positive bits for indications that Q(O) and negative bits that not(Q(O)).