PAC learning, for instance, is fundamentally non-Bayesian. Saying that PAC learning approximates Bayesian inference is the same as saying that Bayesian inference approximates PAC learning. It's not a very meaningful statement.

I looked into PAC learning a bit when Scott Aaronson talked about it on his blog, and came to the following conclusion. 'Instead of saying “PAC-learning and Bayesianism are two different useful formalisms for reasoning about learning and prediction” I think we can keep just Bayesianism and reinterpret PAC-learning results as Bayesian-learning results which say that in some special circumstances, it doesn’t matter exactly what prior one uses. In those circumstances, Bayesianism will work regardless.'

Of course that was 7 years ago and I probably barely scratched the surface of the PAC learning literature even then. Are there any PAC learning results which can't be reinterpreted this way?