What is Bayes' theorem?
Answers
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Bayes' theorem (also known as Bayes' rule) is a useful tool for calculating conditional probabilities. Bayes' theorem can be stated as follows:
Bayes' theorem.. Let A1, A2, ... , An 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( Ak ∩ B ) = P( Ak )P( B | Ak ), Baye's 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 ) ]
Unless you are a world-class statiscian, Bayes' theorem (as expressed above) can be intimidating. However, it really is easy to use. The remainder of this lesson covers material that can help you understand when and how to apply Bayes' theorem effectively.
When to Apply Bayes' Theorem
Part of the challenge in applying Bayes' theorem involves recognizing the types of problems that warrant its use. You should consider Bayes' theorem when the following conditions exist.
The sample space is partitioned into a set of mutually exclusive events { A1, A2, . . . , An }.
Within the sample space, there exists an event B, for which P(B) > 0.
The analytical goal is to compute a conditional probability of the form: P( Ak | B ).
You know at least one of the two sets of probabilities described below.
P( Ak ∩ B ) for each Ak
P( Ak ) and P( B | Ak ) for each Ak
Formula: P(Ei/A) = P(Ei). P(A/Ei)
P(A)
= P(Ei). P(A/Ei)
∑(i=1)P(Ei). P(A/Ei)