Allow me to suggest a simpler thought experiment, that hopefully captures the essence of yours, and shows why your interpretation (of the correct math) is incorrect.
There are 100 recording studios, each recording each day with probability 0.5. Everybody knows that.
There’s a red light outside each studio to signal that a session is taking place that day, except for one rogue studio, where the signal is reversed, being off when there’s a session and on when there isn’t. Only persons B and C know that.
A, B and C are standing at the door of a studio, but only C knows that it’s the rogue one. How do their beliefs that there’s a session inside change by observing that the red light is on? A keeps the 50-50. B now thinks it’s 99-1. Only C knows that there’s no session.
So your interpretation, as I understand it, would be to say that A and B updated in the “wrong direction”. But wait! I practically gave you the same prior information that C has—of course you agree with her! Let’s rewrite the last paragraph:
A, B and C are standing at the door of a studio. For some obscure reason, C secretly believes that it’s the rogue one. Wouldn’t you now agree with B?
And now I can do the same for A, by not revealing to you, the reader, the significance of the red lights. My point is that as long as someone runs a Bayesian update, you can’t call that the “wrong direction”. Maybe they now believe in things that you judge less likely, based on the information that you have, but that doesn’t make you right and them wrong. Reality makes them right or wrong, unfortunately there’s no one around who knows reality in any other way than through their subjective information-revealing observations.
Allow me to suggest a simpler thought experiment, that hopefully captures the essence of yours, and shows why your interpretation (of the correct math) is incorrect.
There are 100 recording studios, each recording each day with probability 0.5. Everybody knows that.
There’s a red light outside each studio to signal that a session is taking place that day, except for one rogue studio, where the signal is reversed, being off when there’s a session and on when there isn’t. Only persons B and C know that.
A, B and C are standing at the door of a studio, but only C knows that it’s the rogue one. How do their beliefs that there’s a session inside change by observing that the red light is on? A keeps the 50-50. B now thinks it’s 99-1. Only C knows that there’s no session.
So your interpretation, as I understand it, would be to say that A and B updated in the “wrong direction”. But wait! I practically gave you the same prior information that C has—of course you agree with her! Let’s rewrite the last paragraph:
A, B and C are standing at the door of a studio. For some obscure reason, C secretly believes that it’s the rogue one. Wouldn’t you now agree with B?
And now I can do the same for A, by not revealing to you, the reader, the significance of the red lights. My point is that as long as someone runs a Bayesian update, you can’t call that the “wrong direction”. Maybe they now believe in things that you judge less likely, based on the information that you have, but that doesn’t make you right and them wrong. Reality makes them right or wrong, unfortunately there’s no one around who knows reality in any other way than through their subjective information-revealing observations.
No matter what, someone is still updating in the wrong direction, even if we don’t know who it is.