section_path: Mathematical treatment/Inverse probability
section_name: Inverse probability
paragraph_text: Inverse probability
  
  In probability theory and applications, Bayes' rule relates the odds of event A 1 {\displaystyle A_{1}} to event A 2, {\displaystyle A_{2},} before (prior to) and after (posterior to) conditioning on another event B. {\displaystyle B.} The odds on A 1 {\displaystyle A_{1}} to event A 2 {\displaystyle A_{2}} is simply the ratio of the probabilities of the two events. When arbitrarily many events A {\displaystyle A} are of interest, not just two, the rule can be rephrased as posterior is proportional to prior times likelihood, P (A | B) ∝ P (A) P (B | A) {\displaystyle P(A|B)\propto P(A)P(B|A)} where the proportionality symbol means that the left hand side is proportional to (i.e., equals a constant times) the right hand side as A {\displaystyle A} varies, for fixed or given B {\displaystyle B} (Lee, 2012; Bertsch McGrayne, 2012). In this form it goes back to Laplace (1774) and to Cournot (1843); see Fienberg (2005).