There’re a couple things he seemed to gloss over, but those seemed more like “boilerplate that was ‘more of the same’ for certain bits, IIRC” rather than “significant things that we’re missing significant bits of”… but then, I guess “glossing over” is a problem because it makes things seem like that. :)
Anyways, I happen to be a fan of vulnerability/coherence/dutch book style arguments. I mean, for cleanliness/simplicity, those just win hands down. (a touch of, at most, linear algebra vs the functional analysis of Cox’s Theorem? :)) And in some forms build up decision theory right at the same time!
Although, now I’m wondering… just how much weaker is Cox’s theorem than Jaynes makes it sound?
The proof in Jaynes applies proves that if you want to assign plausibilities to propositions, and those plausibilities are going to be real numbers, and P(a^b|c) is a function of P(a|b^c) and p(b|c) and P(not a) is a function of P(a) and all those functions are twice-differentiable, and P satisfies a certain density requirement, then P has to be isomorphic to probability.
It just doesn’t have the same philosophical punch as “a few criteria that just seem like common sense that everyone should agree are desirable in a reasoning system.” when you actually spell out the assumptions and they contain seemingly unjustified technical things like differentiability and density.
There are a bunch of rigorous variations on Cox’s theorem, but as far as i know there is nothing that lives up to the hype.
Well, some of those criteria at least seem perfectly reasonable.
As far as what which thing was a function of, IIRC, he kinda went through that, discussing some examples and basically outlining an argument for what sort of things could depend on what vs what would lead to absurdities, so the “this is a function of this and that” wasn’t, IIRC, pulled out of thin air.
There’re a couple things he seemed to gloss over, but those seemed more like “boilerplate that was ‘more of the same’ for certain bits, IIRC” rather than “significant things that we’re missing significant bits of”… but then, I guess “glossing over” is a problem because it makes things seem like that. :)
Anyways, I happen to be a fan of vulnerability/coherence/dutch book style arguments. I mean, for cleanliness/simplicity, those just win hands down. (a touch of, at most, linear algebra vs the functional analysis of Cox’s Theorem? :)) And in some forms build up decision theory right at the same time!
Although, now I’m wondering… just how much weaker is Cox’s theorem than Jaynes makes it sound?
The proof in Jaynes applies proves that if you want to assign plausibilities to propositions, and those plausibilities are going to be real numbers, and P(a^b|c) is a function of P(a|b^c) and p(b|c) and P(not a) is a function of P(a) and all those functions are twice-differentiable, and P satisfies a certain density requirement, then P has to be isomorphic to probability.
It just doesn’t have the same philosophical punch as “a few criteria that just seem like common sense that everyone should agree are desirable in a reasoning system.” when you actually spell out the assumptions and they contain seemingly unjustified technical things like differentiability and density.
There are a bunch of rigorous variations on Cox’s theorem, but as far as i know there is nothing that lives up to the hype.
Well, some of those criteria at least seem perfectly reasonable.
As far as what which thing was a function of, IIRC, he kinda went through that, discussing some examples and basically outlining an argument for what sort of things could depend on what vs what would lead to absurdities, so the “this is a function of this and that” wasn’t, IIRC, pulled out of thin air.