Kurros

Keynes in his "Treatise on probability" talks a lot about analogies in the sense you use it here, particularly in "part 3: induction and analogy". You might find it interesting.

Hmm, thanks. Seems similar to my description above, though as far as I can tell it doesn't deal with my criticisms. It is rather evasive when it comes to the question of what status models have in Bayesian calculations.

That sounds to me more like an argument for needing lower p-values, not higher ones. If there are many confounding factors, you need a higher threshold of evidence for claiming that you are seeing a real effect.

Physicists need low p-values for a different reason, namely that they do very large numbers of statistical tests. If you choose p=0.05 as your threshold then it means that you are going to be claiming a false detection at least one time in twenty (roughly speaking), so if physicists did this they would be claiming false detections every other day and their credibility would plummet like a rock.

Is there any more straightforward way to see the problem? I argued with you about this for a while and I think you convinced me, but it is still a little foggy. If there is a consistency problem, surely this means that we must be vulnerable to Dutch books doesn't it? I.e. they would not seem to be Dutch books to us, with our limited resources, but a superior intelligence would know that they were and would use them to con us out of utility. Do you know of some argument like this?

Very well, then i will wait for the next entry. But i thought the fact that we were explicitly discussing things the robot could not compute made it clear that resources were limited. There is clearly no such thing as logical uncertainty to the magic logic god of the idealised case.

No we aren't, we're discussing a robot with finite resources. I obviously agree that an omnipotent god of logic can skip these problems.

It was your example, not mine. But you made the contradictory postulate that P("wet outside"|"rain")=1 follows from the robots prior knowledge and the probability axioms, and simultaneously that the robot was unable to compute this. To correct this I alter the robots probabilities such that P("wet outside"|"rain")=0.5 until such time as it has obtained a proof that "rain" correlates 100% with "wet outside". Of course the axioms don't determine this; it is part of the robots prior, which is not determined by any axioms.

You haven't convinced nor shown me that this violates Cox's theorem. I admit I have not tried to follow the proof of this theorem myself, but my understanding was that the requirement you speak of is that the probabilistic logic reproduces classical logic in the limit of certainty. Here, the robot is not in the limit of certainty because it cannot compute the required proof. So we should not expect to get the classical logic until updating on the proof and achieving said certainty.

You haven't been very specific about what you think I'm doing incorrectly so it is kind of hard to figure out what you are objecting to. I corrected your example to what I think it should be so that it satisfies the product rule; where's the problem? How do you propose that the robot can possibly set P("wet outside"|"rain")=1 when it can't do the calculation?