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Unformatted text preview: Bayes’ Rule Bayes’ Rule  Updating Probabilities • Let A 1 ,…, A k be a set of events that partition a sample space such that (mutually exclusive and exhaustive): – each set has known P(A i ) > 0 (each event can occur) – for any 2 sets A i and A j , P(A i and A j ) = 0 (events are disjoint) – P(A 1 ) + … + P(A k ) = 1 (each outcome belongs to one of events) • If C is an event such that – 0 < P(C) < 1 ( C can occur, but will not necessarily occur) – We know the probability will occur given each event A i : P(CA i ) • Then we can compute probability of A i given C occurred: ) ( ) and ( ) ( )  ( ) ( )  ( ) ( )  ( )  ( 1 1 C P C A P A P A C P A P A C P A P A C P C A P i k k i i i = + + = Example  OJ Simpson Trial • Given Information on Blood Test (T+/T) – Sensitivity: P(T+Guilty)=1 – Specificity: P(TInnocent)=.9957 ⇒ P(T+I)=.0043 • Suppose you have a prior belief of guilt: P(G)=p* • What is “posterior” probability of guilt after seeing evidence that blood matches: P(GT+)? 0043 . * 9957 . * ) 0043 *)(. 1 ( ) 1 ( * ) 1 ( * ) ( )  ( ) ( ) ( ) ( )  ( ) 0043 *)(. 1 ( ) 1 ( * )  ( ) ( )  ( ) ( ) ( ) ( ) ( + = + = = = + = = + = + = + + + + + + + + + + p p p p p T P G T P G P T P G T P T G P p p I T P I P G T P G P I T P G T...
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This note was uploaded on 06/04/2011 for the course STA 4321 taught by Professor Staff during the Spring '08 term at University of Florida.
 Spring '08
 Staff
 Mutually Exclusive, Probability

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