I am confused, I thought we were to weight hypotheses by 2^-(kolmogorov(H)) not 2^-length(H). Am I missing something?

I'm going by what I've read of Jaynes, Yudkowsky, and books by a couple of other writers on Bayesian statistics.

I don't believe there are any legitimate issues with Bayesian statistics, because Bayes's rule is derived from basic desiderata of rationality which I find entirely convincing, and it seems to me that the maximum entropy principle is the best computable approximation to Solomonoff induction (although I'd appreciate other opinions on that).

There may be legitimate issues with people failing to apply the simple mathematical laws of probability theory correctly, because the correct application can get very complicated - but that is not an issue with Bayesian statistics per se. I'm sure that in many cases, the wisest thing to do might be to use frequentist methods, but being a Bayesian does not prohibit someone from applying frequentist methods when they are a convenient approximation.

I think the interpretation of probability and what methods to use for inference are two separate debates. There was a really good discussion post on this a while back.

I'm also curious as to who exactly these frequentists are that you are arguing against. Perhaps I am spoiled by hanging out with people who regularly have to solve statistical problems, and therefore need to have a reasonable conception of statistics, but most frequentist sentiments that I encounter are fairly well-reasoned, sometimes even pointing out legitimate issues with Bayesian statistics. It is true that I sometimes get incorrect claims that I have to correct, but I don't think becoming a Bayesian magically protects you from this.

EDIT: To clarify, the "frequentist sentiments" I referred to did not explicitly distinguish between interpretations of probability and inference algorithms, but as the goal was engineering I think the arguments were all implicitly pragmatic.

It already exists, if poorly maintained.

Welcome!

I want to work on the hard questions of philosophy, grue and induction, cognition and consciousness, nominalism v.s. realism, Bayesian epistemology, philosophy of probability and mathematics in general, and even meta-physics, though I would like to positivize the field a bit.

That's a huge amount of philosophy to look at. Might I suggest narrowing your interests down a bit, at least at first? It's very easy to read a little bit of everything, but much harder to contribute something non-trivial to every field.

Lastly, I would like to ask, how does less wrong see itself? I mean what is the general LW opinion of what LW is? Is it a blog? An open source research institute? A philosophical movement? A non-philosophical movement? A self-help movement? I am curious.

It seems to be a little bit of all of those things. Some people here are rabidly anti-philosophy, and so if LW overtly called itself a philosophical movement, those people would probably end up evaporating off. On the other hand, some people would very much like to see the self-help aspects of LW become secondary to the more philosophical or technical aspects. Like everything else, it's a bit hard to pin down to a distinct category.

So what if anything is the standard lesswrong approach to Nelson Goodman's grue problem? If there is any paradox I could imagine someone posing against LW, I would imagine it would be the Grue problem.

(damn down voters edit): Not that I think it would pose any real threat. Just curious, I'm sure LW has a brilliant solution. And if not it can def be made by assembling the bits of other posts. I would really like to know why this got down voted.

"Lesswrongianism" however, that's a badass word.

It would be badderass in a dead language. "Minorifalsianism" or something.

Welcome!

I want to work on the hard questions of philosophy, grue and induction, cognition and consciousness, nominalism v.s. realism, Bayesian epistemology, philosophy of probability and mathematics in general, and even meta-physics, though I would like to positivize the field a bit.

That's a huge amount of philosophy to look at. Might I suggest narrowing your interests down a bit, at least at first? It's very easy to read a little bit of everything, but much harder to contribute something non-trivial to every field.

Lastly, I would like to ask, how does less wrong see itself? I mean what is the general LW opinion of what LW is? Is it a blog? An open source research institute? A philosophical movement? A non-philosophical movement? A self-help movement? I am curious.

It seems to be a little bit of all of those things. Some people here are rabidly anti-philosophy, and so if LW overtly called itself a philosophical movement, those people would probably end up evaporating off. On the other hand, some people would very much like to see the self-help aspects of LW become secondary to the more philosophical or technical aspects. Like everything else, it's a bit hard to pin down to a distinct category.

Wouldn't the rule be something more like:

((P(H|E) > P(H)) if and only if (P(H) > P(H|~E))) and ((P(H|E) = P(H)) if and only if (P(H) = P(H|~E)))

So, if some statement is evidence of a hypothesis, its negation must be evidence against. And if some statement's truth value is independent of a hypothesis, then so is that statements negation.

This is implied by the expectation of posterior probabilities version. Since P(E) + P(~E) = 1, that means that P(H|E) and P(H|~E) are either equal, or one is greater than P(H) and one is less than. If they were both less than P(H), then P(H|E)P(E)+P(H|~E)P(~E) would have a lesser value than the largest conditional probability in that formula; suppose P(H|E) is the greater one, then P(H|E)P(E)+P(H|~E)P(~E) < P(H|E) and P(H|E) < P(H), so P(H|E)P(E)+P(H|~E)P(~E) ≠ P(H). If they are both larger than P(H), then P(H|E)P(E)+P(H|~E)P(~E) must be larger than the smallest conditional probability in that formula; suppose that P(H|E) is the smaller one, then we have P(H|E)P(E)+P(H|~E)P(~E) > P(H|E), and P(H|E) > P(H), so P(H) ≠ P(H|E)P(E)+P(H|~E)P(~E). And if both posterior probabilities are equal, then P(H|E)P(E)+P(H|~E)P(~E) = P(H|E), and both posteriors must eqaul the prior. Q.e.d.

I think that the formula that expresses the prior as the average of the posterior probability weighted by the probabilities of observing that evidence and not observing that evidence, is a great way to express the point of this article. But it might not be trivial for everyone to get:

((P(H|E) > P(H)) if and only if (P(H) > P(H|~E))) and ((P(H|E) = P(H)) if and only if (P(H) = P(H|~E)))

from

P(H) = P(H|E)P(E) + P(H|~E)P(~E)

That something is evidence in favor if and only if its negation is evidence against, and that some result is independent of some hypothesis if and only if not observing that result is independent of that hypothesis, are the take home messages of this post as far as i can tell. The law that "P(H) = P(H|E)P(E) + P(H|~E)P(~E)" says more than that, it also tells you how to get P(H|~E) from P(H|E), P(H) and P(E). But adding the boolean statement and its proof from the weighted average statement to the post, or at least to a comment on this post, not even necessarily using the boolean symbols or formalisms, might help a lot of students that come across this long after algebra class. I know it would have helped me.