L20_ECE4001_Fall_2009

L20_ECE4001_Fall_2009 - Lecture 20 Bayesian Decision Making...

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J. Stevenson Kenney © 2009 1 Lecture 20 Bayesian Decision Making And Cost Benefit Analysis November 5, 10, 2009

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J. Stevenson Kenney © 2009 2 Using probability for decision making Probability trees Decision tree methods Can combine with engineering economy Chapter 9 of Hyman
J. Stevenson Kenney © 2009 3 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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J. Stevenson Kenney © 2009 4 Evaluating Risk • A = Circuit works • B = Circuit fails • If P(A) > P(B), choose A • If P(A) < P(B), choose B • Note that P(A) + P(B) = 1 A B • When confronted with multiple options, one may need to assess the risk of each option • Simple example: P(A) P(B)
J. Stevenson Kenney © 2009 5 Dependencies • A and B must be occur before C or D can be chosen A B C D C D P(A) P(B) P(D|A) P(C|A) P(D|B) P(C|B) Note: P(A) + P(B) = 1 P(C|A) + P(D|A) = 1 P(C|B) + P(D|B) = 1

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J. Stevenson Kenney © 2009 6 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
J. Stevenson Kenney © 2009 7 “Inverting” the Decision Tree • We may need to prepare for later decisions before making earlier decisions – Planning resource allocation – Purchasing materials – Scheduling services • To do this, we must “invert” the decision tree using Bayes’ theorem

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J. Stevenson Kenney © 2009 8 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
J. Stevenson Kenney © 2009 9 Bayes’ Theorem ( | ) the probability of event given event has occurred may be calculated using: () (| ) ) where, a set of mutually exclusive and mutually exhaustive events: ( ) 1 a d ii i i i i PA B A B PAPB A PB AP A B = = == = ependent event that occurs only if event occurs ( ) the probability of event occuring ( | ) the probability of event given event has occurred ( ) the probability of event occuring i A PA A PB A B A B = = =

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J. Stevenson Kenney © 2009 10 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 Forward Tree is “reactive” (little planning) Reverse Tree is “responsive” (much planning)

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L20_ECE4001_Fall_2009 - Lecture 20 Bayesian Decision Making...

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