Improbable simple hypotheses are unbelievable

Intuitively, it seems that any simple hypothesis should receive a reasonable prior probability. It seems unreasonable to assign a simple hypothesis a ridiculously low probability. In some sense this is a matter of preference, but here I’ll briefly argue:

If H is a simple hypothesis and the prior probability of H is very small, then there is nothing you could possibly see which would convince you of H.

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Are you special? Pascal’s wager, anthropic reasoning, and decision theory

Here is an argument that some people might find compelling:

It may be that the world is mad, and that as the only sane person around it falls on me to make sure we don’t all kill ourselves. If that’s the case, then my impact on the world may be huge. Let’s say that in this case, I can improve the world by 1%.

Maybe the claim that I’m particularly influential, call it proposition P, isn’t certain. But at least there’s a good chance. ¬†Subjectively it feels like about 1%, since if I looked at 100 similarly surprising facts, I would expect one of them to be true. (I wouldn’t be¬†that¬†surprised to discover that I’m the most important person ever…) That still leaves me with the ability to improve the world by 0.01% in expectation, which looks pretty good. I might as well not even worry about stuff I could do that would improve the world by a mere 0.001%, like being an extraordinarily successful entrepreneur.

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