The big problem with interval arithmetic, stated in terms of probability distributions, isn't that the distributions are uniform, but that the distributions are

The big problem with interval arithmetic, stated in terms of probability distributions, isn't that the distributions are uniform, but that the distributions are independent. The distribution of x-x is the degenerate distribution 0. But the naive implementation of uncertainty propagation would not take into account the dependence between x and x, and would give an over-approximation. It's true that probability distributions generalize sets, and sets generalize intervals. But there are many applications where intervals are a great choice for representing uncertainty. Changing a distribution to uniform is not obviously a bad idea because you can parameterize a uniform distribution and represent the parameters compactly. There is no compact representation for arbitrary distributions.