[nextaccountic]

About the last remark, my intuition is that even though there are operational differences, any formalism to represent uncertainty should be roughly as useful as each other

I mean. Can you express Bayes rule using interval arithmetic? Or something similar to it

[tel]

I think a more complete way to say it would be that probability theory is a refinement of interval theory. Per that last remark, I suspect that if you add any probability measure to intervals such that it has positive weight along the length of the interval then the upper and lower bounds will be preserved.

So in that sense, they're consistent, but interval theory intentionally conveys less information.

Bayes' Law arises from P(X, Y) = P(X | Y)P(Y). It seems to me in interval math, probability downgrades to just a binary measurement of whether or not the interval contains a particular point. So, we can translate it like (x, y) \in (X, Y) iff (y \in Y implies x \in X) and (y \in Y) which still seems meaningful.

[nextaccountic]

Do you know any material that compares probability theory with interval arithmetic like this? I can't find it

[tel]

I don't. I've never actually seen interval theory developed like I did above. It's just me porting parts of probability theory over to solve the same problems as they appear in talking about intervals.