Everybody would presumably agree that it would be better for Alice to consult her gut feelings when she has more evidence rather than less. For each possible future course of events, she should therefore ask herself what subjective probabilities her gut would come up with after experiencing these events. In the likely event that these posterior probabilities turn out to be inconsistent with each other, she should then take account of the confidence she has in her current snap judgements to massage her posterior probabilities until they become consistent. After the massaging is over, Alice would have eliminated the possibility of being surprised by a black-swan event. She would already have taken account of the impact that all future information might have on the unformalized internal model that she uses in determining her beliefs. She would not only have adjusted the subjective probabilities she attaches to events in the small world with which she begins, but potentially expanded her state space to a larger small world in light of the new possibilities that black-swan events commonly suggest (Gillies 2001; Williamson 2003).
The end-product of Alice’s massaging process is therefore a bunch of consistent posteriors defined on a state space that she will never need to revise. With the consistency axioms of Savage’s theory of subjective probability, her massaged posteriors can all be deduced by Bayes’ rule from a single prior. In this story, Bayes’ rule is therefore reduced to a mere book-keeping tool that saves Alice from having to remember all her massaged posterior probabilities. The prior that Savage attributes to Alice therefore squeezes all the juice that can be squeezed from the disorderly set of impressions with which she comes to the problem.
In short, the procedure is something like:
Compute the “posteriors” for all possible future evidence (in your case, this is done the Bayesian way, but in general doesn’t have to be).
Notice the inconsistencies in those “posteriors” and ways in which they seem off.
Coherentify/massage those posteriors until they start making sense.
In particular, unconditioning each posterior on its piece of evidence should get you back at the same prior probability.
Interestingly, this is broadly consistent with Leonard Savage’s view of priors, at least according to Ken Binmore https://link.springer.com/article/10.1007/s41412-017-0056-1
In short, the procedure is something like:
Compute the “posteriors” for all possible future evidence (in your case, this is done the Bayesian way, but in general doesn’t have to be).
Notice the inconsistencies in those “posteriors” and ways in which they seem off.
Coherentify/massage those posteriors until they start making sense.
In particular, unconditioning each posterior on its piece of evidence should get you back at the same prior probability.