Bayes's theorem depends on degrees of belief, not on induction, nor on the belief that the future will be like the past. In order for updating not to be valid, you have to exclude the possibility of degrees of belief. In fact, if you read through a proof of Bayes' theorem, you will see that it nowhere assumes the future will be like the past. In the same way, I could write this whole post so that it is entirely in the past or entirely in the future; the similarity of past and future is not relevant.