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DanielLC comments on Probability and radical uncertainty - Less Wrong
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The Bayesian Universalist answer to this would be that there is no separate meta-probability. You have a universal prior over all possible hypotheses, and mutter a bit about Solomonoff induction and AIXI.
I am putting it this way, distancing myself from the concept, because I don't actually believe it, but it is the standard answer to draw out from the LessWrong meme space, and it has not yet been posted in this thread. Is there anyone who can make a better fist of expounding it?
You can give a meta-probability if you want. However, this makes no difference in your final result. If you are 50% certain that a box has a diamond in it with 20% probability, and you are 50% certain that it has a diamond with 30% probability, then you are 50% sure that it has an expected value of 0.2 diamonds and 50% sure that it has an expected value of 0.3 diamonds, so it has an expected expected value of 0.25 diamonds. Why not just be 25% sure from the beginning?
Supposedly, David gave an example of meta-probability being necessary in the earlier post her references. However, using conditional probabilities give you the right answer. There is a difference between a gambling machine having independent 50% chances of giving out two coins when you put in one, and one that has a 50% chance the first time, but has a 100% chance of giving out two coins the nth time given that it did the first time and a 0% chance given it did not. Since there are times where you need conditional probabilities and meta-probabilities won't suffice, you need to have conditional probabilities anyway, so why bother with meta-probabilities?
That's not to say that meta-probabilities can't be useful. If the probability of A depends on B, and all you care about is A, meta-probabilities will model this perfectly, and will be much simpler to use than conditional probabilities. A good example of a successful use of meta-probabilities is Student's t-test, which can be thought of as a distribution of normal distributions, in which the standard deviation itself has a probability distribution.