Bayesianism has no rules for what someone priors should be. It has rules about how to progress from a state of having priors.
There's a reason that Einstein did not get his Nobel Prize for the special theory of relativity. At the time the prize was given, the Nobel Prize committee did not believe that the predictions about Mercuries orbit were strong enough evidence for the special theory of relativity to give him the Nobel Prize for it. Besides Mercuries orbit there was also the Michelson-Morley experiment.
To the extend that you were taught in school that the Michelson-Morley and Mercuries orbit provided definite evidence for the special theory of relativity, that's a retrospective accounting from people who already knew it to be true and not the perspective from the physicists that gave Einstein his Nobel Prize.
Whether or not physicists at the time should have updated more strongly into the direction of seeing special relativity as proven depends a lot about what you believe of the merits of alternative explanations for the observations and how well those fitted the data and how likely you consider the measurement for Mercuries orbit to be correct.
[Unsure] The probability of GR being true is independent of whether the Bayesian knows about it or not
In Bayesanism probabilities are not independent of the model of the observer the same way that frequentism has a notion of observer independent probabilities.
The general mechanism with the mail order scam you talked about is called survivorship bias. If you take the question of how high the existential risk of being nuclear war happens to be, this matters. If you just observe that we now have nuclear weapons for a long time, it might be wrong to update with each passing year into a lower chance of nuclear war because you would not be around to observe reality in case everyone got killed by nuclear war. That's why we need to look at Petrov and Arkhipov to get an understanding about near misses and we treat both of them as heroes on LessWrong.
Can Bayesianism deal with the mail-order prophet scam? If so, how?
This scam uses a segmented mailing list and a large initial population to advertise predictions so that a 'winning' portion of recipients receive an apparently improbable series of correct predictions. Most recipients (the 'losers') receive an incorrect prediction at some point.
For clarity, here is the situation:
A Bayesian with no prior knowledge of the scam receives a letter advertising stock predictions via a new proprietary quantitative model. The letter predicts that by the end of the month, stock XYZ will be up. It also states that the firm will prove their legitimacy by sending 6 more predictions (one per month) that will correctly predict whether XYZ is up or down. Sure enough, the Bayesian receives each letter, and the predictions all end up correct!
Unbeknownst to the Bayesian, 63 other Bayesians also received the first letter, 31 others received the second, 15 the third, 7 the fourth, 3 the fifth, and 1 other received the sixth letter. Our unlucky Bayesian was the only one to receive the seventh.
The question is, assuming no knowledge of the scam or communication with the "losers" (for whom a stock prediction was wrong), should the Bayesian -- strictly adhering to Bayesian Epistemology -- believe that the firm's stock predictions are legit?
I don't know how to approach this from a Bayesian perspective and would appreciate any guidance from someone who's more familiar. My intuition is that the Bayesian should believe that the predictions are legit (which we observers know to be incorrect).
Also I feel I should disclose that I'm not a Bayesian (rather a Critical Fallibilist), and I'm looking for the best Bayesian answers to this problem (links to prior material welcome). It's often quite hard to find the best answers to a problem from a particular school, so please bear with me if I've missed something obvious.