Replying to your edit:

Without induction, you cannot update on evidence, because "evidence" is a hollow concept.

Still wrong. As I pointed out in the main post, "The sun will rise tomorrow" has a probability of 1.0 given that "The sun will rise for the next 10,000 days." This means that the sun rising tomorrow is evidence that the sun will rise for the next 10,000 days, without presupposing induction.

For example, I might originally be convinced that the probability of the sun rising tomorrow is one in a billion, and the probability of it rising for the next 10,000 days, one in a google; i.e. I am convinced that induction is wrong and the future will not be like the past. Nonetheless Bayes' theorem inexorably forces me to update in favor of the sun rising for the next 10,000 days, if it rises tomorrow.