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.
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.