The prior of a hypothesis does not depend on its complexity

In other words, universal prior assigns probability to points, not events. Probability of an event is sum over its elements, which is not related to the complexity of the event specification (and hypotheses are usually events, not points, especially in the context of Bayesian updating, which is why high-complexity hypotheses can have high probability according to universal prior).

For an explicit example of a high-complexity event with high prior, let one-element event S be one of very high Kolmogorov complexity, and therefore contain a single program that is assigned very low universal prior. Then, not-S is an event with about the same K-complexity as S, but which is assigned near-certain probability by universal prior (because it includes all possible programs, except for the one high-complexity program included in S).

I'm new here, so maybe I'm just being stupid, but I'm massively confused by this (and previous discussion hasn't cleared it up for me):

The complexity of "A and B and C and D" is roughly equal to the complexity of "A or B or C or D"

That to me seems to be an utterly unfounded assertion. Of course the complexity of the two statements is more or less the same, but the the complexity of the state of the world given by A&B&C&D would be the length of the shortest description of a state of the world satisfying A and B and C and D, whereas the complexity of A|B|C|D would be the length of the shortest description of a state of the world satisfying at least one of the conditions. The latter one is potentially much smaller than the former, is it not?

Your example 2, "the correct theory of everything is the lexicographically least algorithm with K-complexity 3^^^^3", has the same problem as the "islam is true" example. K-complexity is uncomputable.

However, "the correct theory of everything is the lexicographically least algorithm in with length 3^^^^3" is computable - and the universal prior probability for it is low, but reasonable given the length of the sentence - it's not related to the magnitude of 3^^^^3.

Here at LessWrong, we're used to using "3^^^^3" as a way to point to a generic very big number, and certainly if you replaced "3^^^^3" with an explicit random number of similar magnitude, then the prior would be very low. But 3^^^^3 is different than most numbers of that size - it has a very short description.

If there was evidence suggesting that there was a very large constant somewhere in our physics, and someone showed that the last few digits of that constant matched the last few digits of 3^^^^3, then we would jump towards the hypothesis that in fact, the number is exactly 3^^^^3.

You haven't convinced me that Occam's prior is false.

I would claim that example 1 is an invalid criticism because Occam's prior only claims to work on sets of mutually exclusive hypotheses, example 2 is an invalid criticism because K-complexity is uncomputable, so the quoted hypothesis is outlawed (which may be unsatisfactory, I don't know the solution to that), and example 3 is an invalid criticism because in the case of P(Hypothesis|Prior Information) Occam's prior should only care about the stuff to the left of the "|".

The other problem with disregarding Occam's prior is that it has a proof. The proof is at least good enough to refute claims that there's no connection with complexity at all.

Re-reading your post I think that your definition of hypothesis is wider than mine, and when I say hypothesis I mean what you call a "predictor". If you apply Occam's prior to predictors (instead of the "wide"/"narrow" thing) do our positions become the same?

Good post, but I object to the title. On average, more complex hypotheses must have lower prior probabilities, just for reasons of summability to 1.

I think a more proper title would be "The prior of a hypothesis is not a function of its complexity".

Can you elaborate on how it might defuse Pascal's Mugging? It seems the problem there is that, no matter how low your prior, the mugger can just increase the number of victims until the expected utility of paying up overwhelms that of not paying. Hypothesis complexity doesn't seem to play in, and even if I were using it to assign a low prior, this could still be overcome.

That said, any solution to the problem (Robin's of course being a good start) is more than welcome.

I think your general conclusion is correct: "does subscription to the universal prior imply assigning probability 2^-m to any hypothesis of length/complexity m? .. ahh no". But perhaps obviously so?

The key is the universal prior deals with programs that are complete predictors of an entire sequence of observations. They are not just simple statements.

If you want to compare to simple statements, you need to quantify their predictive power. The universal prior is a secondary classification, a way of ranking a set of algorithms/hypotheses that all are 100% perfectly accurate and specific. Its like a secondary weighting procedure you use when you have many equally perfect choices given your current data. ( given my understanding refreshed by your description)

That clearly can't apply to statements in general, because statements in general do not in general perfectly predict the whole sequence.

For #1, what are A,B,C,D supposed to be? Complete world predictor programs?

Simple statements? If A&B&C&D is a complete predictor, and so is A|B|C|D, then the universal prior will not choose either: it will choose the simplest of A,B,C,D.

Otherwise, if A&B&C&D is required for a full predictor, then A|B|C|D can not be a full predictor. There is no case then where A&B&C&D has less predictive power than A|B|C|D. The former is more specific and thus has strictly more predictive power, but of course yes is less intrinsically probable.

I think this wrong, but I'm not sure how to directly counter it. so just some notes:

1. When are "A and B" and "A or B" both predictors that fit the existing data?

2. If it turns out that the laws of physics are themselves the result of a simple algorithm, wouldn't that seem obvious in retrospect?

3. The added complexity is not about explanation. Your arrival at .5 is based on a ton of evidence about coins.

The complexity of "A&B&C&D" is roughly equal to the complexity of "A||B||C||D", but we know for certain that the former hypothesis can never be more probable than the latter,

I think you are trying to make formal arguments based on informal definitions, leading to confusion. KC refers to program lengths, but if A,B,C are programs, then what is A||B||C? More careful definitions would lead to a more traditional conclusion.

For one way to formalize the A/B/C/D point, imagine you are trying to solve the UCI Census Income problem, where one attempts to predict whether a person's income is greater than 50k based on various other factors. Let A/B/C/D refer to binary predictors such as (is-married?), (is-religious?), (has-college-degree?), and (is-military?). The goal is to find a combination of these that predicts income well. Now it is of course true that (A||B||C||D) is more likely than (A&B&C&D) for a person. But there is no prior reason to believe that the OR-hypothesis has more predictive power than the AND-hypothesis, just as a KC-style analysis would indicate.

I'm not convinced that example 1 works. "A and B and C" and "A or B or C" do have roughly the same complexity. It is then after we look at the content that we then update our estimates based on the logical form of the claims to give P(A and B and C) < P(A or B or C).

Nice. Can you give some examples related to your last two paragraphs?

Cut some now irrelevant nonsense about Pascal's mugging.

You seem to completely misunderstand Kolmogorov prior, confusing hypotheses with events.

Probability of events "A and B and C and D" is sum of Kolmogorov-derived probabilities of every hypothesis implying "A and B and C and D" that's also consistent with data.

Probability of events "A or B or C or D" is sum of Kolmogorov-derived probabilities of every hypothesis implying "A or B or C or D" that's also consistent with data.

Hypotheses in first set won't be more complex than hypotheses in second set - the second set is simply far far larger.

Second probability consists of things like: (1/2)^length("A") + (1/2)^length("B") + (1/2)^length("A and B") + (1/2)^length("A or B") + (1/2)^length("A and C") + ...