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PhilGoetz comments on Case study: abuse of frequentist statistics - Less Wrong
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The central difficulty of Bayesian statistics is the problem of choosing a prior: where did it come from, how is it justified? How can Bayesians ever make objective scientific statements, if all of their methods require an apparently arbitrary choice for a prior?
Frequentist statistics is the attempt to do probabilistic inference without using a prior. So, for example, the U-test Cyan linked to above makes a statement about whether two data sets could be drawn from the same distribution, without having to assume anything about what the distribution actually is.
That's my understanding, anyway - I would also be happy to see a "Frequentism for Bayesians" post.
A prior gives you as much information as the mean of a distribution. So, can't I by the same token accuse both frequentist and Bayesian statistics of attempting to do probabilistic inference without using a distribution?
I mean, the frequentist uses the U-test to ask whether 2 data sets could be drawn from the same distribution, without assuming what the mean of the distribution is. The Bayesian would use some other test, assuming a prior or perhaps a mean for the distribution, but not assuming a shape for the distribution. And some other, uninvented, and (by the standards of LW) superior statistical methodology would use another test, assuming a mean and a shape for the distribution.
No, not in general, it can give much more or much less; it depends entirely on how detailed you can make your prior. Expanding out e.g. as a series of central moments can give you as detailed a shape as you want. It may reduce to knowing only the mean in certain very special inference problems. In other problems, you may know that the distribution is very definitely Cauchy (EDIT: which doesn't even have a well-defined mean), but not know the parameters, and put some reasonable prior on them -- flat for the center over some range, and approximately using a (1/x) improper prior for the width, perhaps cutting it off at physically relevant length scales.
All that information can be encoded in the prior. The prior covers your probabilities over the space of your hypotheses, not a direct probabilistic encoding of what you think one sample will be.