Bayes Theorem The REVERSE probability theorem The General
Bayes’ Theorem The “REVERSE” probability theorem
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:
Definition (Baptism) A Partition of a sample space S is a finite collection of mutually exclusive, exhaustive events, that is: A picture might help:
The partition of a sample space Some examples will also help
Examples of partitions • (Biology): C 1= Healthy Specimens • C 2= Moderately ill Specimens • C 3= Terminally ill Specimens • C 4= Dead Specimens • (Research): C 1= Control Group (no treatment) • C 2= Monthly Treatment Group • C 3= Weekly Treatment Group • (Manufacture): C 1= Made in USA • C 2= Made in Canada • C 3= Made in China • More Examples will come later.
The next ingredient The next component is some event A which, generally, cuts across the N chunks, as shown in the picture:
The final ingredient • The final component for Bayes’ Theorem are the “givens. ” They are (must be) • P(C 1), P(C 2), …, P(CN) AND …. • P(A|C 1), P(A|C 2), …, P(A|CN) • must be given. We look at examples for the event A in the three situations considered before:
• (Biology): • • Event A = C 1= Healthy Specimens C 2= Moderately ill Specimens C 3= Terminally ill Specimens C 4= Dead Specimens Specimen’s bone weight ≤ 37. 3 kg • (Research): C 1= Control Group (no treatment) • C 2= Monthly Treatment Group • C 3= Weekly Treatment Group • Event A = Subject exhibited marked bone weight gain • (Manufacture): C 1= Made in USA • C 2= Made in Canada • C 3= Made in China • Event A = Item is defective
Bayes’ Theorem has two statements (some authors do not list the first one as part of theorem) Now the proof
In most applications the verbiage may be complex and confusing. My advice is: First determine what, and how many, the chunks are. Then determine the event A Finally, compute the “givens” and apply theorem. We’ll do some examples at the blackboard.
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