How to correctly combine equally weighted predictions and results with uncertainty I understand there have been a lot of answers on combining predictions, but I cannot seem to find anything that answers all of my questions. 
I have 5 regions of the heart to test, and each region can be described on a 1-5 scale (normal = 1 abnormal = 5). Each test I apply tells me, with around 70% \pm 40% confidence (each region is different so these numbers vary a fair bit) whether the particular region is either a 1, 2, 3, 4 or 5. 
How do I combine these predictions to get a confidence of the health of the heart as a whole if:
a) all the regions return the same result ( 5 tests each with a confidence of ~ 70%, all indicating the heart is a healthy 1) 
b) The regions return 2 or more integers (Here i would expect to be able to say that I have a confidence of >70% that the heart is at 1.5, if the two integers returned were 1 and two. 
any help, or redirection to other reading would be greatly appreciated! I'll be on all day trying to figure this out, so if something isn't clear, please leave a comment and I'll respond promptly! 
Thanks.
 A: Some initial comments: How to combine them seems to depend on the use you want to make of this knowledge. This is essential even before trying to find a combined probability. Consider an example different from yours but with the same abstract problem: we know the health conditions of a person's organs and would like to say something about the health condition of the person as a whole. Suppose this person has a liver in very bad condition but a healthy heart (unlikely, but anyway...). Would you say that the person as a whole is healthy? They probably need treatment – they could die because of their liver, even if the heart is healthy. So here the minimum among the statuses of all components could be taken as an indication of the global health. In other situations an average may be more appropriate. Also, is any of the regions more important than the others for the global functioning of the heart?
More formally:
If I understood correctly you want (1) to describe the global health condition by a single number, and (2) to assign a probability to this number. Then I see it this way:
1) The global state of the heart is at first given by a 5-tuple of quantities, each of which can assume five values. That is, your "sample space" is $\{1,\dotsc,5\}^5$. I'd start by writing down a multivariate probability distribution for this 5-tuple. And the question is:
– Are the probabilities of the five quantities independent? In this case the joint probability is just their product.
– Are they not independent? In this case you'd need to find a joint probability that correctly marginalizes to the five marginals. You could check maximum-entropy methods to do this, or assume some other model.
2) Once you have this joint probability, you must find a way to get a single number from the 5-tuple. There are again various ways to do this, eg:
the minimum, the maximum, the average, a weighted average, etc. Each of these defines a function from the space of 5-tuples, $\{1,\dotsc,5\}^5$, to the space $\{1,\dotsc,5\}$. Once you have decided upon the right function, you can find the probability for this number by the usual probability-mapping formulae between different sample spaces.
Sorry for the vagueness of this comment, it seems the answer depends a lot on the context.
