"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 $P (H)$ , while the posterior after seeing the evidence $e$ is often being denoted by the conditional probability $P (H ∣ e) .$ ^[1]

This however is a heuristic rather than a law, and might be false inside some complicated problems. If we've already seen

e_{0}

and are now updating on

e_{1}

, then in this new problem the new prior will be

P (H ∣ e_{0})

and the new posterior will be

P (H ∣ e_{1} \land e_{0}) .

For questions about how priors are "ultimately" determined, see Solomonoff induction.

^{^︎}
E. T. Jaynes was known to insist on using the explicit notation $P (H ∣ I_{0})$ to denote the prior probability of $H$ , with $I_{0}$ denoting the prior, and never trying to write any entirely unconditional probability $P (X)$ . Since, said Jaynes, we always have some prior information.