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AlexMennen comments on An Intuitive Explanation of Solomonoff Induction - Less Wrong
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That doesn't work. If the probability of a regular soldier being able to hit the helmet once is p, then conditional on him hitting the helmet the first time, the probability that he can hit it again is still p. Your argument is the Gambler's fallacy.
(Offtopic: Whoa, what's with the instant downvote? That was a valid point he was making IMO.)
While it is indeed the Gambler's fallacy to believe it to be less than p-likely that he hits me on the second try, I was initially .85 confident that he was non-sniper, which went to .4 as soon as the helmet was hit once. If that was a non-sniper, there was p chances that he hit me, but if it was a sniper, there was s chances that he did.
Once the helmet is hit twice, there is now p^2 odds of this same exact event (helmet-hit-twice) happening, even if the second shot was only p-likely to occur regardless of the result of the first shot. By comparison, the odds that s^2 happens are not nearly as low, which makes the difference between the two possible events greater than the difference between the two possibilities when (helmet-hit-once), hence the greater update in beliefs.
Again, I'm just trying to rationalize on the assumption that the OP intended this to be accurate. There might well be something the author didn't mention, or it could just be a rough approximation and no actual math/bayesian reasoning went into formulating the example, which in my opinion would be fairly excusable considering the usefulness of this article already.
If you update by a ratio s/p for one hit on the helmet, you should update by s^2/p^2 for two hits, which looks just like updating by s/p twice, since updating is just like multiplying by the Bayes factor.
Hmm. I'll have to learn a bit more about the actual theory and math behind Bayes' theorem before I can really go deeper than this in my analysis without spouting out stuff that I don't even know if it's true. My intuitive understanding is that there's a mathematical process that would perfectly explain the discrepancy with minimal unnecessary assumptions, but that's just intuition.
For now, I'll simply update my beliefs according to the evidence you're giving me / confidence that you're right vs confidence (or lack thereof) in my own understanding of the situation.
The interesting bit is that the helmet was hit twice, so we're looking at the probability of being shot twice, not the probability of being shot the second time conditional on being shot the first time.
In retrospect, my first attempt at explanation was fairly poor. Is this clearer?
Edit: To more specifically address your objection: P(hit twice) = P(hit 1st time) * P(hit 2nd time | hit 1st time) = P(hit)^2.
yes