I have just rediscovered an article by Max Albert on my hard drive which I never got around to reading that might interest others on Less Wrong. You can find the article here. It is an argument against Bayesianism and for Critical Rationalism (of Karl Popper fame).
Abstract:
Economists claim that principles of rationality are normative principles. Nevertheless,
they go on to explain why it is in a person’s own interest to be rational. If this were true,
being rational itself would be a means to an end, and rationality could be interpreted in
a non-normative or naturalistic way. The alternative is not attractive: if the only argument
in favor of principles of rationality were their intrinsic appeal, a commitment to
rationality would be irrational, making the notion of rationality self-defeating. A comprehensive
conception of rationality should recommend itself: it should be rational to be
rational. Moreover, since rational action requires rational beliefs concerning means-ends
relations, a naturalistic conception of rationality has to cover rational belief formation including
the belief that it is rational to be rational. The paper considers four conceptions
of rationality and asks whether they can deliver the goods: Bayesianism, perfect rationality
(just in case that it differs from Bayesianism), ecological rationality (as a version of
bounded rationality), and critical rationality, the conception of rationality characterizing
critical rationalism.
Any thoughts?
If we ignore theories with 'maybes', which don't really matter because one theory that predicts two possibilities can easily be split into two theories, weighted by the probabilities assigned by the first theory, then Bayes' theorem simplifies to 'eliminate the theories contradicted by the evidence and rescale the others so the probabilities sum to 1', which is a wonderful way to think about it intuitively. That and a prior is really all there is.
The Solomonoff prior is really just a from of the principle of insufficient reason, which states that if there is no reason to think that one thing is more probable than another, they should be assigned the same probability. Since there are an infinite number of theories, we need to take some kind of limit. If we encode them as self-delimiting computer programs, we can write them as strings of digits (usually binary). Start with some maximum length and increase it toward infinity. Some programs will proceed normally, looping infinitely or encountering a stop instruction, making many programs equivalent because changing bits that are never used by the hypothesis does not change the theory. Other programs will leave the bounds of the maximum length, but this will be fixed as that length is taken to infinity.
This obviously isn't a complete justification, but it is better than Popperian induction. Both rule out falsified theories and both penalize theories for unfalsifiability and complexity. Only Solomonoff induction allows us to quantify the size of these penalties in terms of probability. Popper would agree that a simpler theory, being compared to a more complex one, is more likely but not guaranteed to be true, but he could not give the numbers.
If you are still worried about the foundational issues of the Solomonoff prior, I'll answer your questions, but it would be better if you asked me again in however long progress takes (that was supposed to sound humourous, as if I were describing a specific, known amount of time, but I really doubt that that is noticable in text). http://lesswrong.com/r/discussion/lw/534/where_does_uncertainty_come_from/ writes up some of the questions I'm thinking about now. It's not by me, but Paul seems to wonder about the same issues. This should all be significantly more solid once some of these questions are answered.
"If we ignore theories with 'maybes', which don't really matter because one theory that predicts two possibilities can easily be split into two theories, weighted by the probabilities assigned by the first theory, then Bayes' theorem simplifies to 'eliminate the theories contradicted by the evidence and rescale the others so the probabilities sum to 1', which is a wonderful way to think about it intuitively. That and a prior is really all there is."
That's it? That is trivial, and doesn't solve the major problems in epistemology. It's correct enou... (read more)