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Jonii comments on Pascal's Muggle: Infinitesimal Priors and Strong Evidence - Less Wrong
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I don't at all think that this is central to the problem, but I do think you're equating "bits" of sensory data with "bits" of evidence far too easily. There is no law of probability theory that forbids you from assigning probability 1/3^^^3 to the next bit in your input stream being a zero -- so as far as probability theory is concerned, there is nothing wrong with receiving only one input bit and as a result ending up believing a hypothesis that you assigned probability 1/3^^^3 before.
Similarly, probability theory allows you to assign prior probability 1/3^^^3 to seeing the blue hole in the sky, and therefore believing the mugger after seeing it happen anyway. This may not be a good thing to do on other principles, but probability theory does not forbid it. ETA: In particular, if you feel between a rock and a bad place in terms of possible solutions to Pascal's Muggle, then you can at least consider assigning probabilities this way even if it doesn't normally seem like a good idea.
Actually, there is such a law. You cannot reasonably start, when you are born into this world, naked, without any sensory experiences, expecting that the next bit you experience is much more likely to be 1 rather than 0. If you encounter one hundred zillion bits and they all are 1, you still wouldn't assign 1/3^^^3 probability to next bit you see being 0, if you're rational enough.
Of course, this is mudded by the fact that you're not born into this world without priors and all kinds of stuff that weights on your shoulders. Evolution has done billions of years worth of R&D on your priors, to get them straight. However, the gap these evolution-set priors would have to cross to get even close to that absurd 1/3^^^3... It's a theoretical possibility that's by no stretch a realistic one.