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)

References

[a1]	R.S. Liptser, A.N. Shiryaev, "Statistics of random processes" , 1 , Springer (1977) pp. Section 7.9 (Translated from Russian)