“When the facts change, I change my opinion. What do you do, sir?”
— John Maynard Keynes
In many situations, we don’t follow Keynes’ approach. In fact, in light of new evidence, we usually don’t update our initial beliefs as much as we should. Bayesian inference can help.
Specifically, Bayesian inference allows you to revise the likelihood of a hypothesis h (the prior) in light of a new item of evidence to get to a posterior. The posterior P(h|d) equals your prior P(h) times the conditional probability of the datum of evidence P(d|h) given the hypothesis divided by the probability of the evidence P(d). That is,