# 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.

• I sug­gest mak­ing it ex­plicit that $$P$$ is a dis­tri­bu­tion over a (pos­si­bly in­finite) set of vari­ables (or propo­si­tions nam­ing sym­bols, or what­ever your preferred for­mal­iza­tion is), and that $$P(x)$$ is short­hand for $$P(X=x)$$ when $$X$$ is un­am­bigu­ous. This is one of those things that I had to figure out my­self, which had con­fused me his­tor­i­cally in my youth, and led me to think that all the $$P$$ no­ta­tion was prob­a­bly in­for­mal ar­gu­ment rather than for­mal math.