Jaynes' $A_p$ distribution 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’."
 A: The main point of $A_p$ theory is to describe how cumulative knowledge changes the probability distributions of new observations, given a model.
Starting from a full ignorance model, consistent observations increase the strength of a belief (narrow variance of $(A_p|E)$), while ignorance or inconsistency generate larger instability of the belief (greater variance of $(A_p|E)$.
By introducing $A_p$, Jayes:

*

*shows that a "brain" only needs to store $(A_p|E)$ to make future predictions

*gives a model of inductive reasoning

*underlines the importance of Laplace's rule of succession

Jaynes argues that the $A_p$ densities are the core idea to describe how human brain summarizes previous experience into models (inner robot). The inner robot continuously encodes experience updating $A_p$ and combining it into complex forms can be the basis of creativity (Chapter 18.4).
In practice, it is like you said. The difference between the $A_p$ for the coin and Mars examples are captured by the parameters of the $\text{Beta}$ distributions, but the concept is deeper than that.
