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> You have one variable which stands for p (distributed according to some prior, like a Beta distribution perhaps) This prior sounds like the A_p distribution.
by philh 11y ago
> You have one variable which stands for p (distributed according to some prior, like a Beta distribution perhaps)
This prior sounds like the A_p distribution.
The point is it's a distribution, not a number. You can't just say "I think the probability of seeing heads is p". But if you say "I think the probability of seeing heads is distributed like this, which has a mean of p", then that captures everything you need to know about the coin to update your expectations when you see the results.
(In case it was unclear, A_p isn't a specific shape of distribution, or class of shapes of distributions, like beta or gaussian. It's a distribution over hypothesis-space, where the hypotheses are probabilities, and can have any valid distribution shape.)
- bonoboTP 11y agoI was suggesting that this is less "mysterious" than it seems to sound, once you use a hierarchical model, where one random variable's outcome is used as the parameter for another variable. Imagine one random variable p at the top center of a page, and then several variables under it, each connected by an arrow coming from p. The random variables down there are the coin flips (each results in 0 or 1), and they are identically and independently distributed with a Bernoulli(p) distribution. And p itself is a random variable which has a distribution (expressing how likely each p is). I'm just clarifying that all this fits into normal probability theory, there is no need to introduce any sort of "meta-probability theory".