P(sane things plus crazy things | speaker is saner) P(speaker is saner) = P(speaker is saner | sane things plus crazy things) P(sane things plus crazy things)
The fact that P(sane things plus crazy things | speaker is saner) <> P(speaker is saner | sane things plus crazy things) isn't a problem, if you deal with your priors correctly.
I think I misinterpreted your original question as meaning "Why is this problem fundamentally difficult even for Bayesians?", when it was actually, "What's wrong with the reasoning used by the speaker in addressing this problem?"
Reply to: Shalmanese's Third Law
From an unpublished story confronting Vinge's Law, written in 2004, as abstracted a bit:
Spot the Bayesian problem, anyone? It's obvious to me today, but not to the me of 2004. Eliezer2004 would have seen the structure of the Bayesian problem the moment I pointed it out to him, but he might not have assigned it the same importance I would without a lot of other background.