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CRP 834: Decision Analysis Week One Notes Jean-Michel Guldmann Sumei Zhang

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Statistical Decision Theory Problem setup: –Two alternatives about the state of nature: A null hypothesis ( ) and an alternative one ( ); Decision rule: –Make decision based on a critical value; Action: –Reject or accept the null hypothesis based on a sample; Type I vs. Type II Error

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Statistical Decision Theory Example: College students’ IQ score follows a normal distribution with mean 125, standard deviation students from OSU make the sample. Their average IQ score is 135.

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Statistical Decision Theory Comments: –No account of the seriousness of the consequences of committing type I and type II errors; –No information about the states of nature; –Choice between two alternatives.

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Decision Rules Under Uncertainty Elements of Decision Making –Problem –Objective –Alternative –Consequences –Tradeoffs –Uncertainty –Risk Tolerance –Interaction Decision Making

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Decision Rules Under Uncertainty Concepts –Loss –Regret –Risk Example –Loss Function Loss = l (State, Action)

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Decision Rules Under Uncertainty –Regret Function –Additional Information: probabilities

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Decision Rules Under Uncertainty –8 Possible Decision Rules

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Decision Rules Under Uncertainty –Risk Function Risk = g [ Loss ] = g [ f (State, Action) ] In case of the example: When the state of nature is W1=Rain: When the state of nature is W2=No Rain:

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Decision Rules Under Uncertainty Decision Rules –Look at the average of the risks –Look at the Expected risk (Bayes Risk)

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Decision Rules Under Uncertainty –Comments about Bayes Risk Incorporates the losses due to committing Type I and Type II errors; Provide room for a policy maker’s subjective evaluation; Evaluation of the Bayes risk can be improved by use of the Bayes theorem;

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Decision Rules Under Uncertainty Example (using Bayes Theorem): Of all applicants for a job, it is felt that 75% are able to do the job, and 25% are not. To aid in the selection process, an aptitude test is designed such that a capable applicant has a probability 0.8 of passing test while an incapable one a probability of 0.4 of passing it. An applicant passes the test—what is the probability that he will be able to do the job?

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Decision Rules Under Uncertainty –More Decision Rules The Maximin criterion The Maximax criterion The Hurwicz criterion The Bayes (Laplace) Criterion The Minimax regret criterion Mixed strategy

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