Prior probability

“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 \(H\) is often being written with the unconditioned notation \(\mathbb P(H)\), while the posterior after seeing the evidence \(e\) is often being denoted by the conditional probability \(\mathbb P(H\mid e).\)noteE. T. Jaynes was known to insist on using the explicit notation \(\mathbb P (H\mid I_0)\) to denote the prior probability of \(H\), with \(I_0\) denoting the prior, and never trying to write any entirely unconditional probability \(\mathbb P(X)\). Since, said Jaynes, we always have some prior information.

knows-requisite(Math 2): 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 \(\mathbb P(H\mid e_0)\) and the new posterior will be \(\mathbb P(H\mid e_1 \wedge e_0).\)

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