They invoke Bayes' theorem for its property of being optimal, but it's optimal (and trivial) when you know the priors. The hard part is knowing those, and they don't, so they just use it in a metaphorical/mindset way that doesn't actually mean anything, pretend they don't have actual beliefs so they don't need to take responsibility for them, and say "prior" a lot. (https://metarationality.com/bayesianism-updating)
In general, using probability theory as a way of thinking doesn't work because it can't reject infinitesimally likely but infinitely bad futures, which is why they're overly concerned with imaginary future AIs torturing them, and it ignores "unknown unknowns" and can only handle enumerated futures you already thought of, which is why they have that odd tendency to start polycules because it seems logical and then their girlfriends unexpectedly leave them.
(That's also how you know it's a Berkeley new religion - weird sex practices.)