I mentioned to Eliezer once a few years ago that a weak form of Occam’s razor, across a countable hypothesis space, is inevitable. This observation was new to him so it seems worth reproducing here.
Suppose P(i) is any prior whatsoever over a countable hypothesis space, for example the space of Turing machines. The condition that ∑iP(i)=1, and in particular the condition that this sum converges, implies that for every ϵ>0 we can find N such that ∑i≥NP(i)<ϵ; in other words, the total probability mass of hypotheses with sufficiently large indices gets arbitrarily small. If the indices index hypotheses by increasing complexity, this implies that the total probability mass of sufficiently complicated hypotheses gets arbitrarily small, no matter what the prior is.
The real kicker is that “complexity” can mean absolutely anything in the argument above; that is, the indexing can be arbitrary and the argument will still apply. And it sort of doesn’t matter; any indexing has the property that it will eventually exhaust all of the sufficiently “simple” hypotheses, according to any other definition of “simplicity,” because there aren’t enough “simple” hypotheses to go around, and so must eventually have the property that the hypotheses being indexed get more and more “complicated,” whatever that means.
So, roughly speaking, weak forms of Occam’s razor are inevitable because there just aren’t as many “simple” hypotheses as “complicated” ones, whatever “simple” and “complicated” mean, so “complicated” hypotheses just can’t have that much probability mass individually. (And in turn the asymmetry between simple and complicated is that simplicity is bounded but complexity isn’t.)
There’s also an anthropic argument for stronger forms of Occam’s razor that I think was featured in a recentish post: worlds in which Occam’s razor doesn’t work are worlds in which intelligent life probably couldn’t have evolved.
worlds in which Occam’s razor doesn’t work are worlds in which intelligent life probably couldn’t have evolved.
Can you elaborate or share a link? I would be suspicious of such an argument. To the contrary, I’d say, humans are very complex and also model the world very well. If there could exist simpler models of comparable accuracy, then human complexity would not have evolved.
a weak form of Occam’s razor, across a countable hypothesis space, is inevitable.
Absolutely true, but it’s a bit of a technicality. Remember that the number of hypotheses you could write down in the lifetime of the universe is finite. A prior on those hypotheses can be arbitrarily un-Occamian.
I mentioned to Eliezer once a few years ago that a weak form of Occam’s razor, across a countable hypothesis space, is inevitable. This observation was new to him so it seems worth reproducing here.
Suppose P(i) is any prior whatsoever over a countable hypothesis space, for example the space of Turing machines. The condition that ∑iP(i)=1, and in particular the condition that this sum converges, implies that for every ϵ>0 we can find N such that ∑i≥NP(i)<ϵ; in other words, the total probability mass of hypotheses with sufficiently large indices gets arbitrarily small. If the indices index hypotheses by increasing complexity, this implies that the total probability mass of sufficiently complicated hypotheses gets arbitrarily small, no matter what the prior is.
The real kicker is that “complexity” can mean absolutely anything in the argument above; that is, the indexing can be arbitrary and the argument will still apply. And it sort of doesn’t matter; any indexing has the property that it will eventually exhaust all of the sufficiently “simple” hypotheses, according to any other definition of “simplicity,” because there aren’t enough “simple” hypotheses to go around, and so must eventually have the property that the hypotheses being indexed get more and more “complicated,” whatever that means.
So, roughly speaking, weak forms of Occam’s razor are inevitable because there just aren’t as many “simple” hypotheses as “complicated” ones, whatever “simple” and “complicated” mean, so “complicated” hypotheses just can’t have that much probability mass individually. (And in turn the asymmetry between simple and complicated is that simplicity is bounded but complexity isn’t.)
There’s also an anthropic argument for stronger forms of Occam’s razor that I think was featured in a recentish post: worlds in which Occam’s razor doesn’t work are worlds in which intelligent life probably couldn’t have evolved.
Can you elaborate or share a link? I would be suspicious of such an argument. To the contrary, I’d say, humans are very complex and also model the world very well. If there could exist simpler models of comparable accuracy, then human complexity would not have evolved.
Absolutely true, but it’s a bit of a technicality. Remember that the number of hypotheses you could write down in the lifetime of the universe is finite. A prior on those hypotheses can be arbitrarily un-Occamian.