In Jaynes' book "Probability Theory: The Logic of Science", Jaynes has a chapter (Ch 18) entitled "The $A_p$ distribution and rule of succession" in which he introduces the idea of $A_p$ distributions, which this passage helps illustrate:
[...] To see this, imagine the effect of getting new information. Suppose we tossed the coin five times and it comes up tails every time. You ask me what’s my probability for heads on the next throw; I’ll still say 1/2. But if you tell me one more fact about Mars, I’m ready to change my probability assignment completely [that there was once life on Mars]. There is something which makes my state of belief very stable in the case of the penny, but very unstable in the case of Mars
This might seem to be a fatal objection to probability theory as logic. Perhaps we need to associate with a proposition not just a single number representing plausibility, but two numbers: one representing the plausibility, and the other how stable it is in the face of new evidence. And so, a kind of two-valued theory would be needed. [...]
He goes on to introduce a new proposition $A_p$ such that $$P(A|A_pE) ≡ p$$
"where E is any additional evidence. If we had to render $A_p$ as a verbal statement, it would come out something like this: $A_p$ $≡$ regardless of anything else you may have been told, the probability of A is p."
I'm trying to see the distinction between the two-number idea ("plausibility, and the other how stable it is in the face of new evidence") with just using the Beta distribution which satisfies those criteria.
Fig 18.2 is very similar to using $\alpha=\beta=100$ (say), whereas for Mars it could be Beta(1/2,1/2) and the state of belief is "very unstable"
The original $A_p$ proposition, above, could be Beta($\alpha,\beta$) for very large $\alpha,\beta$ such that $\alpha$/($\alpha+\beta)=p$. Then no amount of evidence would change the distribution of $p$ and $P(A|A_pE) ≡ p$
Beta distribution is discussed throughout the book, so am I missing something that the distinction here is subtle and warranting a new theory ($A_p$ distribution)? He does mention in the very next paragraph "It seems almost as if we are talking about the ‘probability of a probability’."
