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Psy-Kosh comments on Case study: abuse of frequentist statistics - Less Wrong
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Someone correct me if I'm wrong here, but I don't think even having a strong prior P(H) against the evidence is much help, because that makes your likelihood ratio on the evidence P(E|H)/P(E|~H) that much stronger.
(This issue is one my stumbling blocks in Bayescraft.)
The likelihood ratio P(E|H)/P(E|~H) is entirely independent of the prior P(H)
Or did I misunderstand what you said?
In theory, yes, but we're talking about a purported "unswayable Bayesian". If someone strongly believes leprechauns don't exist (low P(H), where H is "leprechauns exist" ), they should strongly expect not to see evidence of leprechauns (low P(E|~H), where E is direct evidence of leprechauns, like finding one in the forest), which suggests a high likelihood ratio P(E|H)/P(E|~H).
I remember Eliezer Yudkowsky referring to typical conversations that go like:
Non-rationalist: "I don't think there will ever be an artificial general intelligence, because my religion says that can't happen."
EY: "So if I showed you one, that means you'd leave your religion?"
He did mention pulling that off once, but I don't believe he said it was typical.
Thanks, that was what I had in mind.
I'm not entirely sure I understand your point. The example you're citing is more the guy saying "I believe X, and X implies ~Y, therefore ~Y", so Eliezer is saying "So Y implies ~X then?"
But the "X implies ~Y" belief can happen when one has low belief in X or high belief in X.
Or are you saying "the likelihoods assigned led to past interpretation of analogous (lack of) evidence, and that's why the current prior is what it is?
komponisto nailed the intuition I was going from: the likelihood ratio is independent of the prior, but an unswayable Bayesian fixes P(E), forcing extreme priors to have extreme likelihood ratios.
*blinks* I think I'm extra confused. The law of conservation of probability is basically just saying that the change in belief may be large or small, so evidence may be strong or weak in that sense. But that doesn't leave the likelihoods up for grabs, (well, okay, P(E|~H) could depend on how you distribute your belief over the space of hypotheses other than H, but... I'm not sure that was your point)
Okay, point conceded ... that still doesn't generate a result that matches the intuition I had. I need to spend more time on this to figure out what assumptions I'm relying on to claim that "extremely wrong beliefs force quick updates".
P(E) = P(E|H) P(H) + P(E|~H)P(~H)
The quantities P(H), P(E|H), and P(E|~H) are in general independent of each other, in the sense that you can move any one of them without changing the others -- provided you adjust P(E) accordingly.
Thanks, that helps. See how I apply that point in my reply to Psy-Kosh here.