I just realized that you may have misunderstood my original point completely. Otherwise you wouldn’t have said this: “I thought the salient feature of Islam was that many people believed it, not that it has less complexity than I thought, or more evidence in its favor than I thought.”
I only used the idea of complexity because that was komponisto’s criterion for the low probability of such claims. The basic idea is people believe things that their priors say do not have too low a probability: but as I showed in the post on Occam’s razor, everyone’s prior is a kind of simplicity prior, even if they are not all identical (nor necessarily particularly related to information theory or whatever.)
Basically, a probability is determined by the prior and by the evidence that it is updated according to. The only reason things are more probable if people believe them is that a person’s belief indicates that there is some human prior according to which the thing is not too improbable, and some evidence and way of updating that can give the thing a reasonable probability. So other people’s beliefs are evidence for us only because they stand in for the other people’s priors and evidence. So it’s not that it is “important that many people believe” apart from the factors that give it probability: the belief is just a sign that those factors are there.
Going back the distinction you didn’t like, between a fixed probability device and a real world claim, a fixed probability device would be a situation where the prior and the evidence is completely fixed and known: with the example I used before, let there be a lottery that has a known probability of one in a trillion. Then since the prior and the evidence are already known, the probability is still one in a trillion, even if someone says he is definitely going to win it.
In a real world claim, on the other hand, the priors are not well known, and the evidence is not well known. And if I find out that someone believes it, I immediately know that there are humanly possible priors and evidence that can lead to that belief, which makes it much more probable even for me than it would be otherwise.
If I find out that … I know that … which makes it much more probable that …
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?
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.
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.
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.
I just realized that you may have misunderstood my original point completely. Otherwise you wouldn’t have said this: “I thought the salient feature of Islam was that many people believed it, not that it has less complexity than I thought, or more evidence in its favor than I thought.”
I only used the idea of complexity because that was komponisto’s criterion for the low probability of such claims. The basic idea is people believe things that their priors say do not have too low a probability: but as I showed in the post on Occam’s razor, everyone’s prior is a kind of simplicity prior, even if they are not all identical (nor necessarily particularly related to information theory or whatever.)
Basically, a probability is determined by the prior and by the evidence that it is updated according to. The only reason things are more probable if people believe them is that a person’s belief indicates that there is some human prior according to which the thing is not too improbable, and some evidence and way of updating that can give the thing a reasonable probability. So other people’s beliefs are evidence for us only because they stand in for the other people’s priors and evidence. So it’s not that it is “important that many people believe” apart from the factors that give it probability: the belief is just a sign that those factors are there.
Going back the distinction you didn’t like, between a fixed probability device and a real world claim, a fixed probability device would be a situation where the prior and the evidence is completely fixed and known: with the example I used before, let there be a lottery that has a known probability of one in a trillion. Then since the prior and the evidence are already known, the probability is still one in a trillion, even if someone says he is definitely going to win it.
In a real world claim, on the other hand, the priors are not well known, and the evidence is not well known. And if I find out that someone believes it, I immediately know that there are humanly possible priors and evidence that can lead to that belief, which makes it much more probable even for me than it would be otherwise.
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.