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.
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.