Using predicted probabilities and bayesian inference to update beliefs
Anything that can be done in the classical, frequentist statistical approach, has a Bayesian counterpart.

Beliefs can be described, under certain assumptions, as probability distributions. You can take a given PDF as your prior belief, combining it with the Likelihood function of your data through Bayes theorem, and obtain a posterior distribution describing your updated belief in light of the empirical evidence provided.

Unfortunately I have no information of your problem at hand, on what data you are working with, and their nature.

If you are interested in Bayesian statistical models I suggest you to take a look at an R package called JAGS, that you can use to implement pretty much any Bayesian model, and with ready-to-go MCMC algorithms.

I wish I could be more specific, please provide more details about your problem and your data.