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Sewing-Machine comments on The Importance of Self-Doubt - Less Wrong
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This sounds like you are updating. We have a formula for what happens when you update, and it indeed says that given evidence, something becomes more probable. You are saying that it becomes much more probable. What quantity in Bayes formula seems especially large to you, and why?
What Wei Dai said.
In other words, as I said before, the probability that people believe something shouldn't be that much more than the probability that the thing is true.
What about the conjunction fallacy?
The probability that people will believe a long conjunction is less probable than they will believe one part of the conjunction (because in order to believe both parts, they have to believe each part. In other words, for the same reason the conjunction fallacy is a fallacy.)
The conjunction fallacy is the assignment of a higher probability to some statement of the form A&B than to the statement A. It is well established that for certain kinds of A and B, this happens.
The fallacy in your proof that this cannot happen is that you have misstated what the conjunction fallacy is.
My point in mentioning it is that people committing the fallacy believe a logical impossibility. You can't get much more improbable than a logical impossibility. But the conjunction fallacy experiments demonstrate that is common to believe such things.
Therefore, the improbability of a statement does not imply the improbability of someone believing it. This refutes your contention that "the probability that people believe something shouldn't be that much more than the probability that the thing is true." The possible difference between the two is demonstrably larger than the range of improbabilities that people can intuitively grasp.
I wish I had thought of this.
You said it before, but you didn't defend it.
Wei Dai did, and I defended it by referencing his position.
In that case I am misunderstanding Wei Dai's point. He says that complexity considerations alone can't tell you that probability is small, because complexity appears in the numerator and the denominator. I will need to see more math (which I guess cousin it is taking care of) before understanding and agreeing with this point. But even granting it I don't see how it implies that P(many believe H)/P(H) is for all H less than one billion.