Bayes ’ Theorem

1. Bayes’Theorem The“REVERSE”probability theorem

2. The “General” Situation A sample space S is “broken up” into chunks Well, maybe N chunks, not just 4. This is called a “PARTITION” and the formal definition is:

3. Definition(Baptism) A Partition of a sample space S is a finite collection of mutually exclusive, exhaustive events, that is: A picture might help:

4. The partition of a sample space Some examples will also help

5. Examples of partitions • (Biology): C1= Healthy Specimens • C2= Moderately ill Specimens • C3= Terminally ill Specimens • C4= Dead Specimens • (Research): C1= Control Group (no treatment) • C2= Monthly Treatment Group • C3= Weekly Treatment Group • (Manufacture): C1= Made in USA • C2= Made in Canada • C3= Made in China • More Examples will come later.

6. The next ingredient The next component is some event A which, generally, cuts across the N chunks, as shown in the picture:

7. The final ingredient • The final component for Bayes’ Theorem are the “givens.” They are (must be) • P(C1), P(C2), …, P(CN)AND …. • P(A|C1), P(A|C2), …, P(A|CN) • must be given. We look at examples for the event A in the three situations considered before:

8. (Biology): C1= Healthy Specimens C2= Moderately ill Specimens C3= Terminally ill Specimens C4= Dead Specimens Event A = Specimen’s bone weight ≤ 37.3kg (Research): C1= Control Group (no treatment) C2= Monthly Treatment Group C3= Weekly Treatment Group Event A = Subject exhibited marked bone weight gain (Manufacture): C1= Made in USA C2= Made in Canada C3= Made in China Event A = Item is defective

9. Bayes’ Theorem has two statements(some authors do not list the first one as part of the theorem)Now the proof

12. In most applications the verbiage may be complex and confusing.My advice is:First determine what, and how many, the chunks are.Then determine theevent AFinally, compute the“givens”and apply the theorem.We’ll do some examples at the blackboard.