Why would the mapping between the language the hypotheses are framed in have impact on which statements are most likley to be true? The article mentions that in domains where the correct hypotheses are complex in the proof language the principle tends to be anti-productive. There is no guarantee that the language is well suited to describe the target phenomenon if we are allowed to freely pick the phenomenon to track!
Wouldn't also any finite complexity class only have finitely many hypotheses in it and wouldn't those also be in a finite numbered index in it? The problem only arises for infinite complexity hypotheses. And it could be argued that if the index is a hyperinteger it can still be a valid placing.
With surreal probability it would be no problem to give an equal infinistemal probability to an infinite list of hypotheses.
I think this is relevant: https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)
The approach of the final authors mentioned on the page seems especially interesting to me. I also am interested to note that their result agrees with Jaynes'. Universability seems to be important to all the most productive approaches there.
This essay claims to refute a popularized understanding of Occam's Razor that I myself adhere to. It is confusing me, since I hold this belief at a very deep level that it's difficult for me to examine. Does anyone see any problems in its argument, or does it seem compelling? I specifically feel as though it might be summarizing the relevant Machine Learning research badly, but I'm not very familiar with the field. It also might be failing to give any credit to simplicity as a general heuristic when simplicity succeeds in a specific field, and it's unclear whether such credit would be justified. Finally, my intuition is that situations in nature where there is a steady bias towards growing complexity are more common than the author claims, and that such tendencies are stronger for longer. However, for all of this, I have no clear evidence to back up the ideas in my head, just vague notions that are difficult to examine. I'd appreciate someone else's perspective on this, as mine seems to be distorted.
Essay: http://bruce.edmonds.name/sinti/