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Lumifer comments on Circular belief updating - LessWrong
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That rather depends on whether the Bayesian (usually known as Bob) knows there is a Tamperer (usually known as Mallory) messing around with his evidence.
If the Bayesian does know, he just distrusts all evidence and doesn't move off his prior. But if he does not know, then the Tamperer just pwns him.
I think your objection is kinda covered by the use of the term "Nash equilibrium" in my comment. And even if the universe decides to create a Tamperer with some probability and leave the evidence untouched otherwise, the result should still hold. The term for that kind of situation is "Bayes-Nash equilibrium", I think.
In this case what's special about Bayesians here?
Bob is playing a zero-sum game against Mallory. All Bob's information is filtered/changed/provided by Mallory and Bob knows it. In this situation Bob cannot trust any of this information and so never changes his response or belief.
I don't see any reason to invoke St.Bayes.
The result also applies if Mallory has limited opportunities to change Bob's information, e.g. a 10% chance of successfully changing it. Or you could have any other complicated setup. In such cases Bob's best strategy involves some updating, and the result says that such updating cannot lower Bob's score on average. (If you're wondering why Bob's strategy in a Nash equilibrium must look like Bayesian updating at all, the reason is given by the definition of a proper scoring rule.) In other words, it's still trivial, but not quite as trivial as you say. Also note that if Mallory's options are limited, her best strategy might become pretty complicated.