# 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.

Parents:

• Bayesian reasoning

A probability-theory-based view of the world; a coherent way of changing probabilistic beliefs based on evidence.