Probability notation for Bayes' rule

Bayes’ rule re­lates prior be­lief and the like­li­hood of ev­i­dence to pos­te­rior be­lief.

Th­ese quan­tities are of­ten writ­ten us­ing con­di­tional prob­a­bil­ities:

  • Prior be­lief in the hy­poth­e­sis: \(\mathbb P(H).\)

  • Like­li­hood of ev­i­dence, con­di­tional on the hy­poth­e­sis: \(\mathbb P(e \mid H).\)

  • Pos­te­rior be­lief in hy­poth­e­sis, af­ter see­ing ev­i­dence: \(\mathbb P(H \mid e).\)

For ex­am­ple, Bayes’ rule in the odds form de­scribes the rel­a­tive be­lief in a hy­poth­e­sis \(H_1\) vs an al­ter­na­tive \(H_2,\) given a piece of ev­i­dence \(e,\) as fol­lows:

$$\dfrac{\mathbb P(H_1)}{\mathbb P(H_2)} \times \dfrac{\mathbb P(e \mid H_1)}{\mathbb P(e \mid H_2)} = \dfrac{\mathbb P(H_1\mid e)}{\mathbb P(H_2\mid e)}.$$

Children:

Parents:

  • Bayes' rule

    Bayes’ rule is the core the­o­rem of prob­a­bil­ity the­ory say­ing how to re­vise our be­liefs when we make a new ob­ser­va­tion.