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TruePath comments on What Are Probabilities, Anyway? - Less Wrong

22 Post author: Wei_Dai 11 December 2009 12:25AM

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Comment author: TruePath 12 December 2009 10:14:49PM *  0 points [-]

Your getting yourself in trouble because you assume that puzzling questions must have deep answers when usually the question itself is flawed or misleading. In this case there just seems to be a need for any explanation of the kind you offer nor would be of any use anyway.

These 'explanations' you offer of probability aren't really explaining anything. Certainly we do succesfully use probability to reason about systems that behave in a deterministic classical fashion (rolling dice probably counts). No matter what sort of probability you believe in you have to explain that application. So introducing 'objective' probability merely adds things we need to explain (possible worlds etc..).

The correct approach is to step back and ask what is it that needs explaining. Well probability is really nothing but a fancy way of counting up outcomes. So once we justify describing the world in a probabilistic fashion (even when it's deterministic in some sense) the application of mathematical inference to reformulate that description in more useful ways is untroubling. In other words if it's reasonable to model rolling two six sided dice as being independent uniformly random variables on 1...6 counting up the combinations and saying there is a 1/6 chance of getting a 7 doesn't raise any new difficulties.

So the question just comes down to is it reasonable of us to model the world using random variables?. I mean one might worry that some worlds were deeply 'tricky' in that almost always when it appeared two objects behaved like independent random variables in reality there was some hidden correlation that would eventually pop out to bite you in the ass and then once you'd taken that correlation into account another one would bite you and so on and so on.

But if you think about it for awhile this isn't really so much a question about the nature of the world as it is a purely mathematical question. If we keep factoring out by our best predictions will the remaining unaccounted for variation in outcomes appear to be random, i.e., make modeling it as random variables an accurate way to make predictions? Well that's actually kinda complicated, I have a theorem (well tiny tweak of someone else's theorem plus interpratation) which I believe says that yes indeed it must work this way. I won't go into it here but let me just say one thing to convince you of it's plausibility.

Basically the argument is that things only fail to look random because we notice a more accurate way of predicting their behavior. The only evidence for a sequence of observations failing to be random according to the supposed distribution would be a pattern in the observations not captured by R so would in turn yield a more accurate distribution. So basically the claim is that we can always simply divide up any observable into the part we can predict (i.e. a distribution of outcomes) and the part we can't. Once you mod out by the part you can predict by defintion anything left is totally unpredictable to you (e.g. computable machines) and thus can't detectably fail to look random according to it's distribution since that would be a better prediction.

This isn't rigorous (it's complicatd) but the point is that Randomness is nothing but our inability to make any better predictions