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Jiro comments on 2013 Survey Results - Less Wrong
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Imagine that 1% of the population have high IQs (and will claim so) and 10% of the population are narcissistic, and half of those like to claim they have high IQ. The Bayseian calculation would be P(high IQ|claim high IQ) = P(claim high IQ|high IQ) * P(high IQ) divided by P(claim high IQ|high IQ) * P(high IQ) + P(claim high IQ|narcissism) * P(narcissism) = (1.00 * 0.01) / (1.00 * 0.01 + 0.5 * 0.10) = 1/6.
You can quibble about the exact figures, but private_messaging is correct here. Because narcissism is relatively common, the claim of having high IQ is very weak evidence for having high IQ but very strong evidence for being narcissistic. (Although it's stronger evidence for high IQ in a community where high IQ is more common.)
Indeed, I think you're way overestimating P(claim high IQ|high IQ).
To clarify, it's still as strong of evidence of having high IQ as a statement can be, it is just not strong enough to overcome the low prior.
Then there's the issue that - I do not know about the US but it seems fairly uncommon to have taken a professionally administered IQ test here, whenever you are smart or not. It may be that LW has an unusually high percentage of people who took such a test.