Bayes' Rule & Theorm

Bayes' Rule (Bayes' Theorem)


“Bayesian inference is important because it provides a normative and general-purpose procedure for reasoning under uncertainty.”

Inductive Reasoning: Experimental, Developmental, and Computational Approaches, edited by Aidan Feeney and Evan Heit

Bayesian inference refers to an approach first proposed by Rev. Thomas Bayes (opens in a new tab) (1702-1761), whose rule allows calculating the probability of an event AA upon observing an event BB.

Bayes' Rule

Bayes' rule or Bayes’ theorem relates the conditional and marginal probabilities of events AA and BB (provided that the probability of BB is not equal to zero). More specifically, Bayes' rule allows calculating the conditional probability of event AA given event BB with the inverse conditional probability of event BB given event A.

P(AB)=P(A)×P(BA)P(B)P(A|B) = P(A) \times {{P(B|A)} \over {P(B)}}



is the conditional probability of event AA given event BB. It is also called the "posterior" probability because it depends on knowledge of event BB. This is the probability of interest.


Note that referring to "posterior" should not be interpreted in a temporal sense, i.e., it does not imply a temporal order between events AA and BB.



is the prior probability (or “unconditional” or “marginal” probability) of event A. The unconditional probability P(A) was first called “a priori” by Sir Ronald A. Fisher (opens in a new tab). It is a “prior” probability because it does not consider any information about event B.


is the prior or marginal probability of event BB.


Note that "prior," just like "posterior," does not imply a temporal order.

Likelihood Ratio

P(BA)P(B){{P(B|A)} \over {P(B)}}

is the Bayes factor or likelihood ratio.

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