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Even if you do want to integrate contrary evidence, it can be hard to do so quickly enough to continue a normal conversation, especially if the evidence is quite unexpected.

For example, suppose I have come to believe that in war X, the victory of the Reds against the Greens was always likely to happen. That is, from the first skirmish between the two sides, one could (with only the information about the two sides available at that point in time) confidently bet on the Reds winning. And the Reds did actually win.

If I know nothing of heuristics and biases, and someone counters my assertion about war X by mentioning the Hindsight Bias, then it may take me quite a long time to integrate this new evidence into my model of the world. I will need to think about the epistemic weight of heuristics and biases information, and ask how closely the conditions of tests of bias resemble my own. If my belief about war X depends in part on evidence about what professional historians believe, then I will have to consider the potentially thorny question of how much professional historians are subject to Hindsight Bias. Of course, my subjective probability in the inevitability of Red victory will go down, but the important question of how much it goes down cannot be answered so easily.

So, what we should often do in face-to-face discussion when we get new evidence is say, "please give me some time to integrate that new evidence into my model". This would be taken by many as a concession of defeat, just as would saying "After conditionalization on your evidence I have lowered by credence in P", and so will be hard for people to do in practice.