Using predicted probabilities and bayesian inference to update beliefs I'm currently working on a project to predict the likelihood of an outcome. I'd like to implement a system where the belief the event will happen is updated after running the model on new data. However, I'm having trouble figuring out exactly how to implement that. Any explanations of updating beliefs using Bayes theorem would be helpful.
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.