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A formula with which it is possible to compute a posteriori probabilities of events (or of hypotheses) from a priori probabilities. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b0153801.png" /> be a complete group of incompatible events: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b0153802.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b0153803.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b0153804.png" />. Then the a posteriori probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b0153805.png" /> of event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b0153806.png" /> if given that event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b0153807.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b0153808.png" /> has already occurred may be found by Bayes' formula:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b0153809.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538010.png" /> is the a priori probability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538012.png" /> is the conditional probability of event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538013.png" /> occurring given event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538014.png" /> (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538015.png" />) has taken place. The formula was demonstrated by T. Bayes in 1763.
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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:
  
Formula (*) is a special case of the following abstract variant of Bayes' formula. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538017.png" /> be random elements with values in measurable spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538019.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538020.png" />. Put, for any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538021.png" />,
+
$$ \tag{* }
 +
{\mathsf P} (A _ {i} \mid  B ) = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538022.png" /></td> </tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538024.png" /> is the indicator of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538025.png" />. Then the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538026.png" /> is absolutely continuous with respect to the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538027.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538028.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538030.png" /> is the Radon–Nikodým derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538031.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015380/b01538032.png" />.
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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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,  "Foundations of the theory of probability" , Chelsea, reprint  (1950)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,  "Foundations of the theory of probability" , Chelsea, reprint  (1950)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.S. Liptser,  A.N. Shiryaev,  "Statistics of random processes" , '''1''' , Springer  (1977)  pp. Section 7.9  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.S. Liptser,  A.N. Shiryaev,  "Statistics of random processes" , '''1''' , Springer  (1977)  pp. Section 7.9  (Translated from Russian)</TD></TR></table>

Latest revision as of 10:33, 29 May 2020


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)

Comments

References

[a1] R.S. Liptser, A.N. Shiryaev, "Statistics of random processes" , 1 , Springer (1977) pp. Section 7.9 (Translated from Russian)
How to Cite This Entry:
Bayes formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bayes_formula&oldid=45997
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article