# Posterior probability

“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 $$\mathbb P(H\mid e).$$

Parents:

• Bayesian reasoning

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