Comment author:robzahra
13 April 2009 09:01:50PM
*
22 points
[-]

Why to accept an inductive principle:

Finite agents have to accept an "inductive-ish" principle, because they can't even process the infinitely many consistent theories which are longer than the number of computations they have in which to compute, and therefore they can't even directly consider most of the long theories. Zooming out and viewing from the macro, this is extremely inductive-ish, though it doesn't decide between two fairly short theories, like Christianity versus string theory.

Probabilities over all your hypotheses have to add to 1, and getting an extra bit of info allows you to rule out approximately half of the remaining consistent theories; therefore, your probability of a theory one bit longer being true ought to drop by that ratio. If your language is binary, this has the nice property that you can assign a 1-length hypothesis a probability of 1/2, a 2-length hypothesis a probability of 1/4, ... an n -length hypothesis a probability of 1/(2^n)...and you notice that 1/2+1/4+1/8 + ... + ~= 1. So the scheme fits pretty naturally.

Under various assumptions, an agent does only a constant factor worse using this induction assumption versus any other method, making this seem not only less than arbitrary, but arguably, "universal".

Ultimately, we could be wrong and our universe may not actually obey the Occam Prior. It appears we don't and can't even in principle have a complete response to religionists who are using solipsistic arguments. For example, there could be a demon making these bullet points seem reasonable to your brain, while they are in fact entirely untrue. However, this does not appear to be a good reason not to use Occam's razor.

Related to (2)--you can't assign equal probability greater than 0 to each of the infinite number of theories consistent with your data, and still have your sums converge to 1 (because for any rational number R > 0, the sum of an infinite number of R's will diverge). So, you have to discount some hypotheses relative to others, and induction looks to be the simplest way to do this (One could say of the previous sentence, "meta-occam's razor supports occam's razor"). The burden of proof is on the religionist to propose a plausible alternative mapping, since the Occam mapping appears to satisy the fairly stringent desiderata.

Further to (5), notice that to get the probability sum to converge to 1, and also to assign each of the infinite consistent hypotheses a probability greater than 0, most hypotheses need to have smaller probability than any fixed rational number. In fact, you need more than that, you actually need the probabilities to drop pretty fast, since 1/2 + 1/3 + 1/4 + .... + does not converge. On the other hand, you COULD have certain instances where you switch two theories around in their probability assignments (for example, you could arbitrarily say Christianity was more likely than string theory, even though Christianity is a longer theory), but for most of the theories, with increasing length you MUST drop your probability down towards 0 relatively fast to maintain the desiderata at all. To switch these probabilities only for particular theories you care about, while you also need and want to use the theory on other problems (including normal "common sense" intuitions, which are very well-explained by this framework), and you ALSO need to use it generally on this problem except for a few counter-examples you explicitly hard-code, seems incredibly contrived. You're better off just to go with occam's razor, unless some better alternative can be proposed.

Comment author:Wei_Dai
12 January 2011 01:29:31AM
3 points
[-]

I agree up to the first half of step 6, but I think the conclusion is wrong (or at least not justified from the argument). There are two different principles involved here:

A finite agent must use an "inductive-ish" prior with a finite complexity

One should use the simplest prior. (Occam's Razor)

If every finite agent must use an "inductive-ish" prior, then there is no need to invoke or appeal to Occam's Razor to explain or justify our own inductive tendencies, so Rob's argument actually undercuts Occam's Razor.

If we replace Occam's Razor with the principle that every finite agent must use a prior with finite complexity, then one's prior is just whatever it is, and not necessarily the simplest prior. There is no longer an argument against someone who says their prior assigns a greater weight to Christianity than to string theory. (In the second half of step 6, Rob says that's "contrived", but they could always answer "so what?")

## Comments (60)

Best*22 points [-]Why to accept an inductive principle:

Finite agents

haveto accept an "inductive-ish" principle, because they can't even process the infinitely many consistent theories which are longer than the number of computations they have in which to compute, and therefore they can't even directly consider most of the long theories. Zooming out and viewing from the macro, this is extremely inductive-ish, though it doesn't decide between two fairly short theories, like Christianity versus string theory.Probabilities over all your hypotheses have to add to 1, and getting an extra bit of info allows you to rule out approximately half of the remaining consistent theories; therefore, your probability of a theory one bit longer being true ought to drop by that ratio. If your language is binary, this has the nice property that you can assign a 1-length hypothesis a probability of 1/2, a 2-length hypothesis a probability of 1/4, ... an n -length hypothesis a probability of 1/(2^n)...and you notice that 1/2+1/4+1/8 + ... + ~= 1. So the scheme fits pretty naturally.

Under various assumptions, an agent does only a constant factor worse using this induction assumption versus any other method, making this seem not only less than arbitrary, but arguably, "universal".

Ultimately, we could be wrong and our universe may not actually obey the Occam Prior. It appears we don't and can't even in principle have a complete response to religionists who are using solipsistic arguments. For example, there could be a demon making these bullet points seem reasonable to your brain, while they are in fact entirely untrue. However, this does

notappear to be a good reason not to use Occam's razor.Related to (2)--you

can'tassignequalprobability greater than 0 to each of the infinite number of theories consistent with your data, and still have your sums converge to 1 (because for any rational number R > 0, the sum of an infinite number of R's will diverge). So, you have to discount some hypotheses relative to others, and induction looks to be the simplest way to do this (One could say of the previous sentence, "meta-occam's razor supports occam's razor"). The burden of proof is on the religionist to propose a plausible alternative mapping, since the Occam mapping appears to satisy the fairly stringent desiderata.Further to (5), notice that to get the probability sum to converge to 1, and also to assign each of the infinite consistent hypotheses a probability greater than 0,

mosthypotheses need to have smaller probability than any fixed rational number. In fact, you need more than that, you actually need the probabilities to drop pretty fast, since 1/2 + 1/3 + 1/4 + .... + does not converge. On the other hand, you COULD have certain instances where you switch two theories around in their probability assignments (for example, you could arbitrarily say Christianity was more likely than string theory, even though Christianity is a longer theory), but formostof the theories, with increasing length you MUST drop your probability down towards 0 relatively fast to maintain the desiderata at all. To switch these probabilities only for particular theories you care about, while you also need and want to use the theory on other problems (including normal "common sense" intuitions, which are very well-explained by this framework), and you ALSO need to use it generally onthisproblem except for a few counter-examples you explicitly hard-code, seems incredibly contrived. You're better off just to go with occam's razor, unless some better alternative can be proposed.Rob Zahra

I agree up to the first half of step 6, but I think the conclusion is wrong (or at least not justified from the argument). There are two different principles involved here:

If every finite agent must use an "inductive-ish" prior, then there is no need to invoke or appeal to Occam's Razor to explain or justify our own inductive tendencies, so Rob's argument actually undercuts Occam's Razor.

If we replace Occam's Razor with the principle that every finite agent must use a prior with finite complexity, then one's prior is just whatever it is, and not necessarily the simplest prior. There is no longer an argument against someone who says their prior assigns a greater weight to Christianity than to string theory. (In the second half of step 6, Rob says that's "contrived", but they could always answer "so what?")

Rob, just make it a post.