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ciphergoth comments on GroupThink, Theism ... and the Wiki - Less Wrong
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To a Bayesian, the problem of induction comes down to justifying your priors. If your priors rate an orderly universe as no more likely than a disorderly one, than all the evidence of regularity in the past is no reason to expect regularity in the future - all futures are still equally likely. Only with a prior that weights more orderly universes with a higher probability, as Solomonoff's universal prior does, will you be able to use the past to make predictions.
More than that, surely: inductive inference is also built into Bayes' theorem itself.
Unless the past is useful as a guide to the future, the whole concept of maintaining a model of the world and updating it when new evidence arrives becomes worthless.
As you say, Bayes' theroem isn't useful if you start from a "flat" prior; all posterior probabilities come out the same as prior probabilities, at least if A is in the future and B in the past. But nothing in Bayes' theorem itself says that it has to be useful.