The post says that radical probabilism rejects #3-#5, but also that Jeffrey’s updates is derived from having rigidity (#5), which sounds like a contradiction.
Jeffrey doesn’t see Jeffrey updates as normative! Like Bayesian updates, they’re merely one possible way to update.
This is also part of why Pearl sounds like a critic of Jeffrey when in fact the two largely agree—you have to realize that Jeffrey isn’t advocating Jeffrey updating in a strong way, only using it as a kind of gateway drug to the more general fluid updates.
I don’t get why the proof of conservation of expected evidence is relevant. It seems to assume that not only do I know how I will update, but that the bookie does too, which seems like an odd and overpowered assumption, and feels in contrast with all the things you said about rigidity – why does the bookie get to know how I’ll update?
Hmm. It seems like a proper reply to this would be to step through the argument more carefully—maybe later? But no, the argument doesn’t require either of those. It requires only that you have some expectation about your update, and the bookie knows what that is (which is pretty standard, because in dutch book arguments the bookies generally have access to your beliefs). You might have a very broad distribution over your possible updates, but there will still be an expected value, which is what’s used in the argument.
I didn’t follow the argument that classical bayesians don’t have calibration. I think it’s just saying that classical bayesianism doesn’t have any part for self-reference, and that’s a big deal? I don’t think this means bayesians aren’t calibrated, just that they don’t have calibration as an explicit part of their model.
Like convergence, this is dependent on the prior, so I can’t say that classical Bayesians are never calibrated (although one could potentially prove some pretty strong negative results, as is the case with convergence?). I didn’t really include any argument, I just stated it as a fact.
What I can say is that classical Bayesianism doesn’t give you tools for getting calibrated. How do you construct a prior so that it’ll have a calibration property wrt learning? Classical Bayesianism doesn’t, to my knowledge, talk about this. Hence, by default, I expect most priors to be miscalibrated in practice when grain-of-truth (realizability) doesn’t hold.
For example, I’m not sure whether Solomonoff induction has a calibration property—nor whether it has a convergence property. These strike me as mathematically complex questions. What I do know is that the usual path to prove nice properties for Solomonoff induction doesn’t let you prove either of these things. (IE, we can’t just say “there’s a program in the mixture that’s calibrated/convergent, so....” … whereas logical induction lets you argue calibration and convergence via the relatively simple “there are traders which enforce these properties”)
Jeffrey doesn’t see Jeffrey updates as normative! Like Bayesian updates, they’re merely one possible way to update.
This is also part of why Pearl sounds like a critic of Jeffrey when in fact the two largely agree—you have to realize that Jeffrey isn’t advocating Jeffrey updating in a strong way, only using it as a kind of gateway drug to the more general fluid updates.
Hmm. It seems like a proper reply to this would be to step through the argument more carefully—maybe later? But no, the argument doesn’t require either of those. It requires only that you have some expectation about your update, and the bookie knows what that is (which is pretty standard, because in dutch book arguments the bookies generally have access to your beliefs). You might have a very broad distribution over your possible updates, but there will still be an expected value, which is what’s used in the argument.
Like convergence, this is dependent on the prior, so I can’t say that classical Bayesians are never calibrated (although one could potentially prove some pretty strong negative results, as is the case with convergence?). I didn’t really include any argument, I just stated it as a fact.
What I can say is that classical Bayesianism doesn’t give you tools for getting calibrated. How do you construct a prior so that it’ll have a calibration property wrt learning? Classical Bayesianism doesn’t, to my knowledge, talk about this. Hence, by default, I expect most priors to be miscalibrated in practice when grain-of-truth (realizability) doesn’t hold.
For example, I’m not sure whether Solomonoff induction has a calibration property—nor whether it has a convergence property. These strike me as mathematically complex questions. What I do know is that the usual path to prove nice properties for Solomonoff induction doesn’t let you prove either of these things. (IE, we can’t just say “there’s a program in the mixture that’s calibrated/convergent, so....” … whereas logical induction lets you argue calibration and convergence via the relatively simple “there are traders which enforce these properties”)
Thank you, those points all helped a bunch.
(I feel most resolved on the calibration one. If I think more about the other two and have more questions, I’ll come back and write them.)