The term you will want to use in your Google search is "Bayesian cognitive science". It's a huge field. But the short answer is, yes, the people in that field do assume that the brain does something that can be modelled as keeping and updating a probability distribution according to Bayes' rule. Much of it is computational-level modelling, i.e. rather removed from questions of implementation in the brain. A quick Google search did, however, find some papers on how to implement Bayesian inference in neural networks - though not necessarily linked to the brain. I'm sure some people do the latter sort of thing as well, though.