# Bayes formula

A formula with which it is possible to compute a posteriori probabilities of events (or of hypotheses) from a priori probabilities. Let $A _ {1} \dots A _ {n}$ be a complete group of incompatible events: $\cup A _ {i} = \Omega$, $A _ {i} \Gamma \cap A _ {j} = \emptyset$ if $i \neq j$. Then the a posteriori probability ${\mathsf P} (A _ {i} \mid B)$ of event $A _ {i}$ if given that event $B$ with ${\mathsf P} (B)>0$ has already occurred may be found by Bayes' formula:

$$\tag{* } {\mathsf P} (A _ {i} \mid B ) = \ \frac{ {\mathsf P} (A _ {i} ) {\mathsf P} (B \mid A _ {i} ) }{\sum _ { i=1 } ^ { n } {\mathsf P} (A _ {i} ) {\mathsf P} (B \mid A _ {i} ) } ,$$

where ${\mathsf P} (A _ {i} )$ is the a priori probability of $A _ {i}$, ${\mathsf P} (B \mid A _ {i} )$ is the conditional probability of event $B$ occurring given event $A _ {i}$( with ${\mathsf P} (A _ {i} ) > 0$) has taken place. The formula was demonstrated by T. Bayes in 1763.

Formula (*) is a special case of the following abstract variant of Bayes' formula. Let $\theta$ and $\xi$ be random elements with values in measurable spaces $( \Theta , B _ \Theta )$ and $(X, B _ {X} )$ and let ${\mathsf E} | g ( \theta ) | < \infty$. Put, for any set $A \in F _ \xi = \sigma \{ \omega : {\xi ( \omega ) } \}$,

$$G(A) = \int\limits _ \Omega g ( \theta ( \omega )) {\mathsf E} [I _ {A} ( \omega ) \mid F _ \theta ] ( \omega ) {\mathsf P} (d \omega ),$$

where $F _ \theta = \sigma \{ \omega : {\theta ( \omega ) } \}$ and $I _ {A} ( \omega )$ is the indicator of the set $A$. Then the measure $G$ is absolutely continuous with respect to the measure ${\mathsf P}$( $G \ll {\mathsf P}$) and ${\mathsf E} [g ( \theta ) \mid F _ \xi ] ( \omega ) = (dG / d {\mathsf P} ) ( \omega )$, where $(dG / d {\mathsf P} ) ( \omega )$ is the Radon–Nikodým derivative of $G$ with respect to ${\mathsf P}$.

#### References

 [1] A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1950) (Translated from Russian)