When two hypotheses agree, what evidence does it give?

[mXSCNT]

Suppose we have two hypotheses H1 and H2. We also have a potential conclusion C. We know all prior probabilities involving H1, H2, and C.

Now, we are given new information that H1 -> C and also H2 -> C - they agree on the same conclusion. Let's call this new information A (for Agreement). That is, A is the statement "H1 -> C and H2 -> C."

How should we revise our belief in H1, H2, and C, given A?

 

[Stephen Tashi]

To turn this into a mathematical question, we would need to define "belief". There are systems such as the Dempster-Shafer theory of evidence that might apply, but I can only comment on the approach by ordinary probability theory.

The question assumes that we know the "prior" joint probability distribution of 3 binary random variables F(x,y,z). The variable x has only two possible outcomes, 0 (for "not-H1") or 1 (for "H1"). Likewise y may be 0 or 1, representing "not-H2" or "H2" and z may be 0 or 1, representing "not-C" or "C".

The implication "H1->C" is equivalent to the statement "not-C or H1". So the updated prior is F(x,y,z | (z = 0 or x = 1) and (z = 0 or y=1) ).

Expressing the prior by Bayes rule involves figuring out the joint probability of some convoluted logical statements. For example as one step in figuring out
F(1,0,1|(z=0 or x=1) and (z = 0 or y = 1) ),
we need to compute the probability of the event:
(x = 1 and y =0 and z = 1) and ( (z =0 or x = 1) and (z = 0 or y = 1) ).

I trust that anyone actually interested in working this problem will do such things and show us the answer!