thomblake comments on Open Thread June 2010, Part 4 - Less Wrong

5 Post author: Will_Newsome 19 June 2010 04:34AM

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Comment author: WrongBot 01 July 2010 07:09:32PM 0 points [-]

Instead of using "optimal general prior", I should have said that I was pessimistic about the existence of a standard for evaluating priors (or, more properly, prior probability distributions) that is optimal in all circumstances, if that's any clearer.

Having thought about the problem some more, though, I think my pessimism may have been premature.

A prior probability distribution is nothing more than a weighted set of hypotheses. A perfect Bayesian would consider every possible hypothesis, which is impossible unless hypotheses are countable, and they aren't; the ideal for Bayesian reasoning as I understand it is thus unattainable, but this doesn't mean that there are benefits to be found in moving toward that ideal.

So, perfect Bayesian or not, we have some set of hypotheses which need to be located before we can consider them and assign them a probabilistic weight. Before we acquire any rational evidence at all, there is necessarily only one factor that we can use to distinguish between hypotheses: how hard they are to locate. If it is also true that hypotheses which are easier to locate make more predictions and that hypotheses which make more predictions are more useful (and while I have not seen proofs of these propositions I'm inclined to suspect that they exist), then we are perfectly justified in assigning a probability to a hypothesis based on it's locate-ability.

This reduces the problem of prior probability evaluation to the problem of locate-ability evaluation, to which it seems maxent and its fellows are proposed answers. It's again possible there is no objectively best way to evaluate locate-ability, but I don't yet see a reason for this to be so.

Again, if I've mis-thought or failed to justify a step in my reasoning, please call me on it.

Comment author: thomblake 01 July 2010 07:55:25PM 0 points [-]

It's again possible there is no objectively best way

I'm not sure I'm willing to grant that's impossible in principle. Presumably, you need to find some way of choosing your priors, and some time later you can check your calibration, and you can then evaluate the effectiveness of one method versus another.

If there's any way to determine whether you've won bets in a series, then it's possible to rank methods for choosing the correct bet. And that general principle can continue all the way down. And if there isn't any way of determining whether you've won, then I'd wonder if you're talking about anything at all (weird thought experiments aside).