My viewpoint is that the prior distributions giving weight 1/3 to each of the three hypotheses is different from the one giving weight 1/2 to each of ν0 and ν1, even if their mixture distributions are exactly the same.
That’s pretty unintuitive to me. What does it matter whether we happen to write out our belief state one way or the other? So long as the predictions come out the same, what we do and don’t choose to call our ‘hypotheses’ doesn’t seem particularly relevant for anything?
We made our choice when we settled on M as the prior. Everything past that point just seems like different choices of notation to me? If our induction procedure turned out to be wrong or suboptimal, it’d be because M was a bad prior to pick, not because we happened to write M down in a weird way, right?
I answered in the parallel thread, which is probably going down to the crux now. To add a few more points:
The prior matters for the Solomonoff bound, see Theorem 5. (Tbc., the true value of the prediction error is the same irrespective of the prior, but the bound we can prove differs)
I think different priors have different aesthetics. Choosing a prior because it gives you a nice result (i.e., Solomonoff prior) feels different from choosing it because it’s a priori correct (like the a priori prior in this post). to me, aesthetics matter.
That’s pretty unintuitive to me. What does it matter whether we happen to write out our belief state one way or the other? So long as the predictions come out the same, what we do and don’t choose to call our ‘hypotheses’ doesn’t seem particularly relevant for anything?
We made our choice when we settled on M as the prior. Everything past that point just seems like different choices of notation to me? If our induction procedure turned out to be wrong or suboptimal, it’d be because M was a bad prior to pick, not because we happened to write M down in a weird way, right?
I answered in the parallel thread, which is probably going down to the crux now. To add a few more points:
The prior matters for the Solomonoff bound, see Theorem 5. (Tbc., the true value of the prediction error is the same irrespective of the prior, but the bound we can prove differs)
I think different priors have different aesthetics. Choosing a prior because it gives you a nice result (i.e., Solomonoff prior) feels different from choosing it because it’s a priori correct (like the a priori prior in this post). to me, aesthetics matter.