It’s actually somewhat tricky to establish that the rules of probability apply to the Frequentist meaning of probability. You have to mess around with long run frequencies and infinite limits. Even once that’s done, it hard to make the case that the Frequentist meaning has anything to do with the real world—there are no such thing as infinitely repeatable experiments.
In contrast, a few simple desiderata for “logical reasoning under uncertainty” establish probability theory as the only consistent way to do so that satisfy those criteria. Sure, other criteria may suggest some other way of doing so, but no one has put forward any such reasonable way.
In contrast, a few simple desiderata for “logical reasoning under uncertainty” establish probability theory as the only consistent way to do so that satisfy those criteria. Sure, other criteria may suggest some other way of doing so, but no one has put forward any such reasonable way.
Could Dempster-Shafer theory count? I haven’t seen anyone do a Cox-style derivation of it, but I would guess there’s something analogous in Shafer’s original book.
It’s actually somewhat tricky to establish that the rules of probability apply to the Frequentist meaning of probability. You have to mess around with long run frequencies and infinite limits. Even once that’s done, it hard to make the case that the Frequentist meaning has anything to do with the real world—there are no such thing as infinitely repeatable experiments.
In contrast, a few simple desiderata for “logical reasoning under uncertainty” establish probability theory as the only consistent way to do so that satisfy those criteria. Sure, other criteria may suggest some other way of doing so, but no one has put forward any such reasonable way.
Could Dempster-Shafer theory count? I haven’t seen anyone do a Cox-style derivation of it, but I would guess there’s something analogous in Shafer’s original book.
I would be quite interested in seeing such. Unfortunately I don’t have any time to look for such in the foreseeable future.