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
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.
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.
Decision Rules Under Uncertainty Elements of Decision Making –Problem –Objective –Alternative –Consequences –Tradeoffs –Uncertainty –Risk Tolerance –Interaction Decision Making
Decision Rules Under Uncertainty Concepts –Loss –Regret –Risk Example –Loss Function Loss = l (State, Action)
Decision Rules Under Uncertainty –Regret Function –Additional Information: probabilities
Decision Rules Under Uncertainty –8 Possible Decision Rules
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:
Decision Rules Under Uncertainty Decision Rules –Look at the average of the risks –Look at the Expected risk (Bayes Risk)
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;
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?
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