I'm allowed to believe whatever I want; I'm just not allowed to try to convince you of it unless I have a rational argument.

Isn't this what Bayesianism is all about -- reaching the most likely conclusion in the face of weak or inconclusive evidence? Or am I misunderstanding something?

The best source to look at here is Probability is Subjectively Objective. You cannot (in the bayesian sense) believe whatever you 'want'. There is precisely one set of beliefs to which you are epistemically entitled given your current evidence even though I are obliged to form a different set of beliefs given what I have been exposed to.