Or, does it just sidestep it? Or, is it just completely unrelated to it? I’m having trouble seeing what the connection might be. It seems to me that Bayesians solve it by saying, take:
Your prior credence in a hypothesis, H.
Your prior credence in observing some evidence, E. (Sometimes P(E) is calculated as the sum of conditional probability of E given H and not-H weighted by the prior credence of H and not-H respectively, and sometimes by breaking not-H down into competing hypotheses.)
Your conditional credence of E given H.
If you observe that E happens, then if your posterior credence in H is anything other than what Bayes’ theorem says it is, you’ve reasoned incorrectly. You’re as objectively wrong as the guy who believes ‘P’, ‘if P then Q’, but then does not accept ‘Q’. If you’re a sane reasoner who applied anything like sensible credences to 1.,2., and 3. for the common “Humean examples” of induction (this A is a B, that A is a B, ..., therefore, probably all As are Bs), your credence in H will surely go up.
Granted, Bayesian epistemology might have its own problems, such as problem of the priors – where do our starting credences ultimately come from, are there objectively correct priors or does anything go? - but this is not the same issue as Hume’s problem of induction, is it? I mean, deductive logic is concerned with validity - getting from true premises to true conclusions - but it says nothing about the actual truth values of the premises. Likewise, inductive logic should be about getting from prior credences to posterior credences, but surely it’s no fault of an inductive logic (anymore than in the case of deductive logic) that it doesn’t tell us what the prior credences are supposed to be?
Hopefully this all makes sense. What do epistemologists and philosophers of science think about the relation between Bayesian epistemology and Hume’s problem of induction?