Statistics and Probability Dictionary
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Bayes Rule
Bayes' theorem (also known as Bayes' rule) is a very useful tool for calculating
conditional probabilities
. Bayes' theorem can be stated as follows:
Bayes' theorem. Let A
1, A
2,
... , A
n be a set of mutually exclusive events that together form
the sample space S. Let B be any event from the same sample space, such that
P(B) > 0. Then,
|
P( Ak | B ) =
|
P( Ak ∩ B )
P( A1 ∩ B ) + P( A2 ∩ B ) + . . . + P( An ∩ B )
|
Note: Invoking the fact that P( A
k ∩ B ) = P( A
k )P( B | A
k ), Bayes' theorem can
also be expressed as
|
P( Ak | B ) =
|
P( Ak ) P( B | Ak )
P( A1 ) P( B | A1 ) + P( A2 ) P( B | A2
) + . . . + P( An ) P( B | An )
|