Editing Probability (section)

===Inverse probability===
{{Main|Inverse probability}}
In [[probability theory]] and applications, ''[[Bayes' theorem|Bayes' rule]]'' relates the [[odds]] of event <math>A_1</math> to event <math>A_2,</math> before (prior to) and after (posterior to) [[Conditional probability|conditioning]] on another event <math>B.</math> The odds on <math>A_1</math> to event <math>A_2</math> is simply the ratio of the probabilities of the two events. When arbitrarily many events <math>A</math> are of interest, not just two, the rule can be rephrased as ''posterior is proportional to prior times likelihood'', <math>P(A|B)\propto P(A) P(B|A)</math> where the proportionality symbol means that the left hand side is proportional to (i.e., equals a constant times) the right hand side as <math>A</math> varies, for fixed or given <math>B</math> (Lee, 2012; Bertsch McGrayne, 2012). In this form it goes back to Laplace (1774) and to Cournot (1843); see Fienberg (2005).