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Decision Analysis Decision-Making Under Risk updated 11.06.01

SMU EMIS 5300/7300. NTU SY-521-N. Systems Analysis Methods Dr. Jerrell T. Stracener, SAE Fellow. Decision Analysis Decision-Making Under Risk updated 11.06.01. Decision Making Under Risk Decision Trees Revising Probabilities Based on Sample Information. Decision Trees.

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Decision Analysis Decision-Making Under Risk updated 11.06.01

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  1. SMU EMIS 5300/7300 NTU SY-521-N Systems Analysis Methods Dr. Jerrell T. Stracener, SAE Fellow Decision Analysis Decision-Making Under Risk updated 11.06.01

  2. Decision Making Under Risk • Decision Trees • Revising Probabilities Based on Sample Information

  3. Decision Trees

  4. Revising Probabilities Base on Sample Information

  5. Bayes Theorem Let {B1, B2, ..., Bn) be a set of events forming a partition of the sample space S, where P(Bi)  0, for i = 1, 2, ... , n. Let A be any event of S such that P(A)  0. Then, for k = 1, 2, ..., n,

  6. In a sense, Bayes’ Rule is updating or revising the prior probability P(B) by incorporating the observed information contained within event A into the model.

  7. Example Let us examine the revision of probabilities using Bayes’ rule with sampling information. Determine how to best invest $10,000 for the next year. However there are only two decision alternatives available to the investor. 1. Bonds 2. Stocks Only two states of the investment climate can occur: 1. No growth 2. Rapid growth

  8. STATE OF NATURE No Growth (0.65) Rapid Growth (0.35) DECISIONALTERNATIVE BondsStocks $500-$200 $100$1100 Example There is a 0.65 probability of no growth in the investment climate and 0.35 probability of rapid growth. The payoffs are $500 for a bond investment in a no-growth state, $100 for a bond investment in a rapid-growth state, -$200 for a stock investment in a no-growth state, and a $1100 payoff for a stock investment in a rapid-growth state.

  9. Example The expected monetary value for the bonds decision alternative is EMV(bonds) = $500(0.65) + $100(0.35) = $360 The expected monetary value for the stocks decision alternative is EMV(stocks) = -$200(0.65) + $1100(0.35) = $255

  10. Example Now suppose that the decision-maker has a chance to obtain some information from an economic expert regarding the future state of the investment economy. This expert does not have a perfect record of forecasting, but she has a track record of predicting a no-growth economy about 0.80 of the time when there actually is a no-growth economy. She has been slightly less successful in predicting rapid-growth economies, with a 0.70 probability of success. ACTUAL STATE OF ECONOMY No Growth (s1) Rapid Growth (s2) Forecaster Predicts No Growth (F1)Forecaster Predicts Rapid Growth (F2) 0.800.20 0.300.70

  11. Example Using these condition probabilities, we can revise prior probabilities of the states of the economy using Bayes’ rule. Suppose the forecaster predicts no growth (F1).

  12. STATE OF ECONOMY STATE OF ECONOMY PRIORPROBABILITIES PRIORPROBABILITIES CONDITIONALPROBABILITIES CONDITIONALPROBABILITIES JOINTPROBABILITIES JOINTPROBABILITIES REVISEDPROBABILITIES REVISEDPROBABILITIES No Growth (s1) No Growth (s1) P (s1)=0.65 P (s1)=0.65 P (F1 |s1)=0.80 P (F1 |s1)=0.20 P (F1s1)=0.130 P (F1s1)=0.520 0.520/0.625 = 0.832 0.130/0.375 = 0.347 Rapid Growth (s2) Rapid Growth (s2) P (s2)=0.35 P (s2)=0.35 P (F1 |s2)=0.30 P (F1 |s2)=0.70 P (F1s2)=0.245 P (F1s2)=0.105 0.105/0.625 = 0.168 0.245/0.375 = 0.653 P (F1)=0.375 P (F1)=0.625 Example Revision Based on Forecast of No Growth (F1) Revision Based on Forecast of Rapid Growth (F2)

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