| There's some abuse of poor notation going on in the article. I don't think the author is intending to be confusing through this imprecision, but instead is just faithfully representing the common way people discuss this kind of stuff. But it is confusing. And it is imprecise. (I'll use x below to mean multiplication due to HN's weird formatting rules) Nominally, if we have two intervals A and B we might anticipate there's a difference between AxA and AxB. In normal math we expect this because we use the different letters to indicate the potential for A and B to be different. Another way of saying it is to say that AxA = AxB exactly when A = B. The trick of language with interval math is that people often want to write things like A = (l, h). This is meaningful, the lower and upper bounds of the interval are important descriptors of the interval itself. But let's say that it's also true that B = (l, h). If A = B, then it's definitely true that their lower and upper bounds will coincide, but is the converse true? Is it possible for two intervals to have coincident bounds but still be unequal? What does equality mean now? In probability math, the same issue arises around the concept of a random variable (rv). Two rvs might, when examined individually, appear to be the same. They might have the same distribution, but we are more cautious than that. We reserve the right to also ask things like "are the rvs A and B independent?" or, more generally, "what is the joint distribution of (A, B)?". These questions reinforce the idea that random variables are not equivalent to their (marginal) distributions. That information is a very useful measurement of a rv, but it is still a partial measurement that throws away some information. In particular, when multiple rvs are being considered, marginal distributions fail to capture how the rvs interrelate. We can steal the formal techniques of probability theory and apply them to give a better definition of an interval. Like an rv, we'll define an interval to be a function from some underlying source of uncertainty, i.e. A(w) and B(w). Maybe more intuitively, we'll think of A and B as "partial measurements" of that underlying uncertainty. The "underlying uncertainty" can be a stand in for all the myriad ways that our measurements (or machining work, or particular details of IEEE rounding) go awry, like being just a fraction of a degree off perpendicular to the walls we're measuring to see if that couch will fit. We'll define the lower and upper bounds of these intervals as the minimum and maximum values they take, l(A) = min_w A(w) and u(A) = max_w A(w). Now, when multiplying functions on the same domain, the standard meaning of multiplication is pointwise multiplication: (A x B)(w) = A(w) x B(w)
and so the lower and upper bounds of AxB suddenly have a very complex relationship with the lower and upper bounds of A and B on their own. l(A x B) = min_w A(w) x B(w)
u(A x B) = max_w A(w) x B(w)
So with all this additional formal mechanism, we can recover how pointwise multiplication makes sense. We can also distinguish AxA and AxB as being potentially very different intervals even when l(A) = l(B) and u(A) = u(B).(As a final, very optional note, the thing that makes interval math different from probability theory is that the underlying space of uncertainty is not endowed with a probability measure, so we can only talk about things like min and max. It also seems like we can make the underlying event space much less abstract and just use a sufficiently high-dimensional hypercube.) |
I mean. Can you express Bayes rule using interval arithmetic? Or something similar to it