There's a useful heuristic to solve tricky questions about "truths" and "beliefs": reduce them to questions about decisions and utilities. For example, the Sleeping Beauty problem is very puzzling if you insist on thinking in terms of subjective probabilities, but becomes trivial once you introduce any payoff structure. Maybe we could apply this heuristic here? Believing in one formulation of a theory over a different equivalent formulation isn't likely to win a Bayesian reasoner many dollars, no matter what observations come in.

EY's very attractive intuition that of two theories making the same predictions, one is true and the other ... what? False? Wrong? Well, ... "not quite so true".

"More Wrong". :)

I can think of two circumstances under which two theories would make the same predictions (that is, where they'd systematically make the same predictions, under all possible circumstances under which they could be called upon to do so):

• They are mathematically isomorphic — in this case I would say they are the same theory.
• They contain isomorphic substructures that are responsible for the identical predictions. In this case, the part outside what's needed to actually generate the predictions counts as extra detail, and by the conjunction rule, this reduces the probability of the "outer" hypothesis.

The latter is where collapse vs. MWI falls, and where "we don't know why the fundamental laws of physics are what they are" vs. "God designed the fundamental laws of physics, and we don't know why there's a God" falls, etc.

The tradition in Bayesianism and standard rationality (and logical positivism, for that matter) that the truth of a statement is to be found through its observable consequences.

Since when is that the Bayesian tradition? Citation needed.

Well the second of those things already has very serious problems. See for example Quine's Confirmation Holism. We've know for a long time that our theories are under-determined by our observations and that we need some other way of adjudicating empirically equivalent theories. This was our basis for preferring Special Relativity over Lorentz Ether Theory. Parsimony seems like one important criteria but involves two questions:

1. One man's simple seems like another man's complex. How do you rigorously identify the more parsimonious between two hypotheses. Lots of people thing God is a very simple hypothesis. The most seemingly productive approach that I know of is the algorithmic complexity one that is popular here.

2. Is parsimony important because parsimonious theories are more likely be 'real' or is the issue really one of developing clear and helpful prediction generating devices?

The way the algorithmic probability stuff has been leveraged is by building candidates for universal priors. But this doesn't seem like the right way to do it. Beliefs are about anticipating future experience so they should take the form of 'Sensory experience x will occur at time t" (or something reducible to this). Theories aren't like this. Theories are frameworks that let us take some sensory experience and generate beliefs about our future sensory experiences.

So I'm not sure it makes sense to have beliefs distinguishing empirically identical theories. That seems like a kind of category error- a map-territory confusion. The question is, what do we do with this algorithmic complexity stuff that was so promising. I think we still have good reasons to be thinking cleanly about complicated science- the QM interpretation debate isn't totally irrelevant. But it isn't obvious algorithmic simplicity is what we want out of our theories (nor is it clear that what we want is the same thing other agents might want out of their theories). (ETA: Though of course K-complexity might still be helpful in making predictions between two possible futures that are empirically distinct. For example, we can assign a low probability to finding evidence of a moon landing conspiracy since the theory that would predict discovering such evidence is unparsimonious. But if that is the case, if theories can be ruled improbable on the basis of the structure of the theory alone why can we only do this with empirically distinct theories? Shouldn't all theories be understandable in this way?)

Thanks, your comment is a very clear formulation of the reason why I wrote the post. Probably even better than the post itself.

I'm halfway tempted to write yet another post about complexity (maybe in the discussion area), summarizing all the different positions expressed here in the comments and bringing out the key questions. The last 24 hours have been a very educational experience for me. Or maybe let someone else do it, because I don't want to spam LW.

"Bayes's rule only deals with the fraction of reality-space spanned by a sentence"

Well, that's the thing: reality-space doesn't concern just our observations of the universe. If two different theories make the same predictions about our observations but disagree about which mechanism produces those events we observe, those are two different slices of reality-space.

Yes, apologies. Fixed above.