"Prior probability", "prior odds", or just "prior" refers to a state of belief that obtained before seeing a piece of new evidence. Suppose there are two suspects in a murder, Colonel Mustard and Miss Scarlet. After determining that the victim was poisoned, you think Mustard and Scarlet are respectively 25% and 75% likely to have committed the murder. Before determining that the victim was poisoned, perhaps, you thought Mustard and Scarlet were equally likely to have committed the murder (50% and 50%). 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 prior probability of a hypothesis is often being written with the unconditioned notation , while the posterior after seeing the evidence is often being denoted by the conditional probability [1]
For questions about how priors are "ultimately" determined, see Solomonoff induction.
E. T. Jaynes was known to insist on using the explicit notation to denote the prior probability of , with denoting the prior, and never trying to write any entirely unconditional probability . Since, said Jaynes, we always have some prior information.