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Will_Sawin comments on Confidence levels inside and outside an argument - Less Wrong
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I don't think the other guy counts as a Bayesian.
That's definitely a good approximation of the organizational structure of the human mind of an imperfect Bayesian. You have a human consciousness simulating a Bayesian probability-computer, but the human contains heuristics powerful enough to, in some situations, overrule the Bayesian.
This has nothing to do with arguments, though.