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Dominik comments on An Intuitive Explanation of Solomonoff Induction - Less Wrong
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Let me dramatize your sniper example a bit with this hypothetical scenario.
You read about homeopathy, you realize that the purported theoretical underpinnings sound nonsensical, you learn that all properly conducted clinical trials show that homeopathy is no better than placebo and so estimate that the odds of it being anything but a sham are, charitably, 1 in a million. Then your mother, who is not nearly as rational, tries to convince you to take Sabadil for your allergies, because it has good product reviews. You reluctantly agree to humor her (after all, homeopathy is safe, if useless).
Lo, your hay fever subsides... and now you are stuck trying to recalculate your estimate of homeopathy being more than a sham in the face of the shear impossibility of the sugar pills with not a single active molecule remaining after multiple dilutions having any more effect than a regular sugar pill.
What would you do?
Do a bayesian update.
Additional to the prior, you need:
(A) The probability that you will heal anyway
(B) The probability that you will heal if sabadil works
I have done the math for A=0,5 ; B=0,75 : The result is an update from 1/10^6 to 1,5 * 1/10^6
For A=1/1000 and B=1 the result is 0.000999
The math is explained here: http://lesswrong.com/lw/2b0/bayes_theorem_illustrated_my_way