Reply to 'Mathematical Inconsistency in Solomonoff Induction?'Date: 2020-09-04
This is a reply to this LessWrong post.
I went through the maths in OP and it seems to check out. I think the core inconsistency is that SI implies . I’m going to redo the maths below (breaking it down step-by-step more). curi has which is the same inconsistency given his substitution. I’m not sure we can make that substitution but I also don’t think we need to.
Let X and Y be independent hypotheses for Solomonoff induction.
According to the prior, the non-normalized probability of X (and similarly for Y ) is:
what is the probability of ?
However, by Equation (1) we have:
This must hold for any and all X and Y .
curi considers the case where X and Y are the same length, starting with Equation (4)
curi has slightly different logic and argues which I think is reasonable. His argument means we get . I don’t think those steps are necessary but they are worth mentioning as a difference. I think Equation (8) is enough.
I was curious about what happens when . Let’s assume the following:
so, from Equation (2)
So there’s an inconsistency regardless of whether or not.
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