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Unknowns comments on Justifying Induction - Less Wrong
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Oh, and not to be nitpicky, but if you're relying on any kind of metric (e.g. your vision) to ascertain that the sun does rise tomorrow, you rely on induction. Without induction, there is simply no way of establishing that your observations correlate with anything meaningful. "The sun will rise tomorrow" cannot actually be confirmed without assuming induction; without the evidence confirming their reliability from past experience, our sensory data are meaningless. This is getting into a nightmarish level of abstraction for a relatively simple point, though.
Believing the sun will rise tomorrow with P=10^-9 is not failing to believe in induction. It's making a serious mistake, or being privy to some very interesting evidence. Without induction, no probability estimate is possible, because we simply have no idea what will happen.
I suspect this argument stems from different definitions of "induction."
If you define believing in induction as believing that the future will be like the past, it is possible to believe that the future will not be like the past, and one example of that would be believing that the sun will not rise tomorrow. Similarly, someone could suppose that everything that will happen tomorrow will be totally different from today, and he could still use Bayes' theorem, if he had any probability estimates at all.
You say, "Without induction, no probability estimate is possible, because we simply have no idea what will happen." Probability estimates are subjective degrees of belief, and it may be that there is some process like induction that generates them in a person's mind. But this doesn't mean that he believes, intellectually, that the future will be like the past, nor that he actually uses this claim in coming up with an estimate; as I just pointed out, some claims explicitly deny that induction will continue to work, and some people sometimes believe them (i.e. "The world will end tomorrow!")
In any case, it doesn't matter how a person comes up with his subjective estimates; a prior probability estimate doesn't need to be justified, it just needs to be used. This post was not intended to justify people's priors, but the process of induction as an explicit reasoning process-- which is not used in generating priors.