That is precisely what I was proposing, just he explains it much better of course. Thanks!

In the subsequent article he makes essentially the same argument as Lumifer's point about this having the potential to be turtles all the way down.

In a comment to the first article the author quotes Jaynes as saying:

What are we doing here? It seems almost as if we are talking about the ‘probability of a probability’. Pending a better understanding of what that means, let us adopt a cautious notation that will avoid giving possibly wrong impressions. We are not claiming that P(Ap|E) is a ‘real probability’ in the sense that we have been using that term; it is only a number which is to obey the mathematical rules of probability theory.

The "pending a better understanding of what that means" is also what I've been grappling with. In last week's thread I initially proposed looking at it as the likelihood that I'll find evidence that will make me change my probability estimate, and then I modified that to being how strong the evidence would have to be to make me change my probability estimate. [Aside from just understanding what "probabilities of probabilities" would actually mean, these ways of expressing it make the concept much more universally applicable than the narrow cases that the linked article is referring to.] But is there a better way of understanding it?