"Posterior probability" or "posterior odds" refers our state of belief after seeing a piece of new evidence and doing a Bayesian update. Suppose there are two suspects in a murder, Colonel Mustard and Miss Scarlet. Before determining the victim's cause of death, perhaps you thought Mustard and Scarlet were equally likely to have committed the murder (50% and 50%). After determining that the victim was poisoned, you now think that Mustard and Scarlet are respectively 25% and 75% likely to have committed the murder. In this case, your "prior probability" of Miss Scarlet committing the murder was 50%, and your "posterior probability" after seeing the evidence was 75%. The posterior probability of a hypothesis $~$H$~$ after seeing the evidence $~$e$~$ is often denoted using the conditional probability notation $~$\mathbb P(H\mid e).$~$