L21_ECE4001_Fall_2008

L21_ECE4001_Fall_2008 - Using probability for decision...

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Slide #1 ECE 4001 L21 © 2008 T. Michaels Lecture 21 Bayesian Decision Making And Cost Benefit Analysis Slide #2 ECE 4001 L21 © 2008 T. Michaels Using probability for decision making Use Bayes’ theorem with application to a series of events Decision tree methods Can combine with engineering economy Chapter 9 of Hyman Slide #3 ECE 4001 L21 © 2008 T. Michaels Bayes’ Theorem B B P A B A B P A A P A B A B P A B P A P B A P B A B A P i i i i i i i i i i i event of y probabilit the ) ( occurred has event given event of y probabilit the ) | ( event of y probabilit the ) ( occurs event if only occurs event that dependent a events exhaustive mutually and exclusive mutually of set a where, ) ( ) | ( ) ( ) | ( : using calculated be may occurred has event given event of y probabilit the ) | ( = = = = = = = Slide #4 ECE 4001 L21 © 2008 T. Michaels Probability Trees Graphical representations consists of: 1. Branches: (straight lines) 2. Event or chance nodes (circles) C D E C D E A B
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Slide #5 ECE 4001 L21 © 2008 T. Michaels Probability Tree Starting With Chance Node (A or B) C D E P(A) C D E A B P(B) P(C|A) P(D|A) P(E|A) P(C|B) P(D|B) P(E|B) P(A) + P(B) = 1 P(C|A) + P(D|A) + P(E|A) = 1 P(C|B) + P(D|B) + P(E|B) = 1 Slide #6 ECE 4001 L21 © 2008 T. Michaels Probability Tree Starting With Chance Node (C, D or E) A B P(C) C E P(A|C) P(B|C) D A B P(A|E) P(B|E) A B P(A|D) P(B|D) P(D) P(E) P(C) + P(D) + P(E) = 1 P(A|C) + P(B|C) = 1 P(A|D) + P(B|D) = 1 P(A|E) + P(B|E) = 1 Slide #7 ECE 4001 L21 © 2008 T. Michaels Probability Tree Starting With Chance Node (C, D or E) A B P(C) C E P(A|C) P(B|C) D A B P(A|E) P(B|E) A B P(A|D) P(B|D) P(D) P(E) C D E P(A) C D E A B P(B) P(C|A) P(D|A) P(E|A) P(C|B) P(D|B) P(E|B) Forward Probability Tree Reverse Probability Tree Use Bayes” theorem to invert probability trees Slide #8 ECE 4001 L21 © 2008 T. Michaels There is a plane crash. The plane crash may be located in the mountains, fields, or in the ocean. An initial search failed to locate the crash site. What is the probability that the crash site is located in the mountains? Region Prob. Located Prob. Finding Mountains 0.5 0.7 Fields 0.3 0.8 Ocean 0.2 0.1 Draw the probability tree starting with the region chance node Draw the probability tree starting with the find/lost chance node
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Slide #9 ECE 4001 L21 © 2008 T. Michaels Region Prob. Located Prob. Finding Mountains 0.5 0.7 Fields 0.3 0.8 Ocean 0.2 0.1 Mt. 0.5 Fields 0.3 Ocean 0.2 Find Lost Find Lost Find Lost 0.7 0.8 0.1 Starting Chance Node - Region 0.3 0.2 0.9 Slide #10 ECE 4001 L21 © 2008 T. Michaels Probability of finding crash site: P(Find) = P(Find|Mt) P(Mt) + P(Find|Fields) P(Fields) + P(Find|Ocean) P(Ocean) P(Find) = 0.7 (0.5) + 0.8 (0.3) + 0.1 (0.2) = 0.61 P(Lost) = 1 – P(Find) = 1 – 0.61 = 0.39 or P(Lost) = P(Lost|Mt) P(Mt) + P(Lost|Fields) P(Fields) + P(Lost|Ocean) P(Ocean) P(Lost) = 0.3 (0.5) + 0.2 (0.3) + 0.9 (0.2) = 0.39 Mt. 0.5 Fields 0.3 Ocean 0.2 Find Lost Find Lost Find Lost 0.7 0.8
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This note was uploaded on 06/09/2009 for the course ECE 4001 taught by Professor Frazier during the Fall '09 term at Georgia Tech.

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L21_ECE4001_Fall_2008 - Using probability for decision...

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