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bentarm comments on Open thread, October 2011 - Less Wrong
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The problem with this interpretation is that it renders the initial statement pretty meaningless. Assuming he's decided to give us a centered 50% confidence interval, which is the only one that really makes sense, that means that 25% of his probability distribution over probabilities is more likely than 1/1000, and this part of the probability mass is going to dominate the rest.
For example, if you think there's a 25% chance that the "actual probability" (whatever that means) is 0.01, then your best estimate of the "actual probability" has to be at least 0.004, which is significantly more than 1/1000, and even a 1% chance of it being 0.1 would already be enough to move your best estimate above 0.001, so it's not just that I'm not sure the concept makes sense, it's that the statement gives us basically no information in the only interpretation in which it does make sense.
Suppose you wanted to make a decision that is equally sensible for P values above X, and not sensible for P values below X. Then, knowing that a chunk of the pdf is below or above X is valuable. (If you only care about whether or not the probability is greater than 1e-3; he's suggested there's a less than 50% chance that's the case).
To elaborate a little more: he's answered one of the first questions you would ask to determine someone's pdf for a variable. One isn't enough; we need two (or hopefully more) answers. But it's still a good place to start.