Chap18 statistical decision theory
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This chapter discusses statistical decision theory and making decisions under uncertainty. It covers constructing payoff tables and decision trees to list possible actions and outcomes. It introduces several decision criteria for selecting the best action, including maximin, minimax regret, and expected monetary value. Expected monetary value calculates the weighted average payoff using the probabilities of different outcomes. Bayes' theorem allows revising probabilities based on new sample information. The key goals are describing decision making processes and criteria for evaluating options under uncertainty.
Chap18 statistical decision theory
- 1. Chapter 18 Statistical Decision Theory Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Statistics for Business and Economics 7th Edition Ch. 18-1
- 2. Chapter Goals After completing this chapter, you should be able to: Describe basic features of decision making Construct a payoff table and an opportunity-loss table Define and apply the expected monetary value criterion for decision making Compute the value of sample information Describe utility and attitudes toward risk Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-2
- 3. Steps in Decision Making List Alternative Courses of Action Choices or actions List States of Nature Possible events or outcomes Determine ‘Payoffs’ Associate a Payoff with Each Event/Outcome combination Adopt Decision Criteria Evaluate Criteria for Selecting the Best Course of Action Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-3 18.1
- 4. List Possible Actions or Events Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Payoff Table Decision Tree Two Methods of Listing Ch. 18-4
- 5. Payoff Table Form of a payoff table Mij is the payoff that corresponds to action ai and state of nature sj Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Actions States of nature s1 s2 . . . sH a1 a2 . . . aK M11 M21 . . . MK1 M12 M22 . . . MK2 . . . . . . . . . . . . M1H M2H . . . MKH Ch. 18-5
- 6. Payoff Table Example A payoff table shows actions (alternatives), states of nature, and payoffs Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Investment Choice (Action) Profit in $1,000’s (States of nature) Strong Economy Stable Economy Weak Economy Large factory Average factory Small factory 200 90 40 50 120 30 -120 -30 20 Ch. 18-6
- 7. Decision Tree Example Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Large factory Small factory Average factory Strong Economy Stable Economy Weak Economy Strong Economy Stable Economy Weak Economy Strong Economy Stable Economy Weak Economy Payoffs 200 50 -120 40 30 20 90 120 -30 Ch. 18-7
- 8. Decision Making Overview Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall No probabilities known Probabilities are known Decision Criteria Nonprobabilistic Decision Criteria: Decision rules that can be applied if the probabilities of uncertain events are not known * maximin criterion minimax regret criterion Ch. 18-8 18.2
- 9. The Maximin Criterion Consider K actions a1, a2, . . ., aK and H possible states of nature s1, s2, . . ., sH Let Mij denote the payoff corresponding to the ith action and jth state of nature For each action, find the smallest possible payoff and denote the minimum M1 * where More generally, the smallest possible payoff for action ai is given by Maximin criterion: select the action ai for which the corresponding Mi * is largest (that is, the action with the greatest minimum payoff) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall )M,,M,Min(MM 1H1211 * 1 )M,,M,(MM 1H1211 * i Ch. 18-9
- 10. Maximin Example Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Investment Choice (Alternatives) Profit in $1,000’s (States of Nature) Strong Economy Stable Economy Weak Economy Large factory Average factory Small factory 200 90 40 50 120 30 -120 -30 20 1. Minimum Profit -120 -30 20 The maximin criterion 1. For each option, find the minimum payoff Ch. 18-10
- 11. Maximin Solution Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Investment Choice (Alternatives) Profit in $1,000’s (States of Nature) Strong Economy Stable Economy Weak Economy Large factory Average factory Small factory 200 90 40 50 120 30 -120 -30 20 1. Minimum Profit -120 -30 20 The maximin criterion 1. For each option, find the minimum payoff 2. Choose the option with the greatest minimum payoff 2. Greatest minimum is to choose Small factory (continued) Ch. 18-11
- 12. Regret or Opportunity Loss Suppose that a payoff table is arranged as a rectangular array, with rows corresponding to actions and columns to states of nature If each payoff in the table is subtracted from the largest payoff in its column . . . . . . the resulting array is called a regret table, or opportunity loss table Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-12
- 13. Minimax Regret Criterion Consider the regret table For each row (action), find the maximum regret Minimax regret criterion: Choose the action corresponding to the minimum of the maximum regrets (i.e., the action that produces the smallest possible opportunity loss) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-13
- 14. Opportunity Loss Example Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Investment Choice (Alternatives) Profit in $1,000’s (States of Nature) Strong Economy Stable Economy Weak Economy Large factory Average factory Small factory 200 90 40 50 120 30 -120 -30 20 The choice “Average factory” has payoff 90 for “Strong Economy”. Given “Strong Economy”, the choice of “Large factory” would have given a payoff of 200, or 110 higher. Opportunity loss = 110 for this cell. Opportunity loss (regret) is the difference between an actual payoff for a decision and the optimal payoff for that state of nature Payoff Table Ch. 18-14
- 15. Opportunity Loss Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Investment Choice (Alternatives) Profit in $1,000’s (States of Nature) Strong Economy Stable Economy Weak Economy Large factory Average factory Small factory 200 90 40 50 120 30 -120 -30 20 (continued) Investment Choice (Alternatives) Opportunity Loss in $1,000’s (States of Nature) Strong Economy Stable Economy Weak Economy Large factory Average factory Small factory 0 110 160 70 0 90 140 50 0 Payoff Table Opportunity Loss Table Ch. 18-15
- 16. Minimax Regret Example Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Investment Choice (Alternatives) Opportunity Loss in $1,000’s (States of Nature) Strong Economy Stable Economy Weak Economy Large factory Average factory Small factory 0 110 160 70 0 90 140 50 0 Opportunity Loss Table The minimax regret criterion: 1. For each alternative, find the maximum opportunity loss (or “regret”) 1. Maximum Op. Loss 140 110 160 Ch. 18-16
- 17. Minimax Regret Example Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Investment Choice (Alternatives) Opportunity Loss in $1,000’s (States of Nature) Strong Economy Stable Economy Weak Economy Large factory Average factory Small factory 0 110 160 70 0 90 140 50 0 Opportunity Loss Table The minimax regret criterion: 1. For each alternative, find the maximum opportunity loss (or “regret”) 2. Choose the option with the smallest maximum loss 1. Maximum Op. Loss 140 110 160 2. Smallest maximum loss is to choose Average factory (continued) Ch. 18-17
- 18. Decision Making Overview Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall No probabilities known Probabilities are known Decision Criteria * Probabilistic Decision Criteria: Consider the probabilities of uncertain events and select an alternative to maximize the expected payoff of minimize the expected loss maximize expected monetary value Ch. 18-18 18.3
- 19. Payoff Table Form of a payoff table with probabilities Each state of nature sj has an associated probability Pi Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Actions States of nature s1 (P1) s2 (P2) . . . sH (PH) a1 a2 . . . aK M11 M21 . . . MK1 M12 M22 . . . MK2 . . . . . . . . . . . . M1H M2H . . . MKH Ch. 18-19
- 20. Expected Monetary Value (EMV) Criterion Consider possible actions a1, a2, . . ., aK and H states of nature Let Mij denote the payoff corresponding to the ith action and jth state and Pj the probability of occurrence of the jth state of nature with The expected monetary value of action ai is The Expected Monetary Value Criterion: adopt the action with the largest expected monetary value Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall H 1j ijjiHHi22i11i MPMPMPMP)EMV(a 1P H 1j j Ch. 18-20
- 21. Expected Monetary Value Example The expected monetary value is the weighted average payoff, given specified probabilities for each state of nature Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Investment Choice (Alternatives) Profit in $1,000’s (States of Nature) Strong Economy (.3) Stable Economy (.5) Weak Economy (.2) Large factory Average factory Small factory 200 90 40 50 120 30 -120 -30 20 Suppose these probabilities have been assessed for these states of nature Ch. 18-21
- 22. Expected Monetary Value Solution Example: EMV (Average factory) = 90(.3) + 120(.5) + (-30)(.2) = 81 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Investment Choice (Action) Profit in $1,000’s (States of nature) Strong Economy (.3) Stable Economy (.5) Weak Economy (.2) Large factory Average factory Small factory 200 90 40 50 120 30 -120 -30 20 Expected Values (EMV) 61 81 31 Maximize expected value by choosing Average factory (continued) Payoff Table: Goal: Maximize expected monetary value Ch. 18-22
- 23. Decision Tree Analysis A Decision tree shows a decision problem, beginning with the initial decision and ending will all possible outcomes and payoffs Use a square to denote decision nodes Use a circle to denote uncertain events Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-23
- 24. Add Probabilities and Payoffs Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Large factory Small factory Decision Average factory States of nature Strong Economy Stable Economy Weak Economy Strong Economy Stable Economy Weak Economy Strong Economy Stable Economy Weak Economy (continued) PayoffsProbabilities 200 50 -120 40 30 20 90 120 -30 (.3) (.5) (.2) (.3) (.5) (.2) (.3) (.5) (.2) Ch. 18-24
- 25. Fold Back the Tree Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Large factory Small factory Average factory Strong Economy Stable Economy Weak Economy Strong Economy Stable Economy Weak Economy Strong Economy Stable Economy Weak Economy 200 50 -120 40 30 20 90 120 -30 (.3) (.5) (.2) (.3) (.5) (.2) (.3) (.5) (.2) EMV=200(.3)+50(.5)+(-120)(.2)=61 EMV=90(.3)+120(.5)+(-30)(.2)=81 EMV=40(.3)+30(.5)+20(.2)=31 Ch. 18-25
- 26. Make the Decision Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Large factory Small factory Average factory Strong Economy Stable Economy Weak Economy Strong Economy Stable Economy Weak Economy Strong Economy Stable Economy Weak Economy 200 50 -120 40 30 20 90 120 -30 (.3) (.5) (.2) (.3) (.5) (.2) (.3) (.5) (.2) EV=61 EV=81 EV=31 Maximum EMV=81 Ch. 18-26
- 27. Bayes’ Theorem Let s1, s2, . . ., sH be H mutually exclusive and collectively exhaustive events, corresponding to the H states of nature of a decision problem Let A be some other event. Denote the conditional probability that si will occur, given that A occurs, by P(si|A) , and the probability of A , given si , by P(A|si) Bayes’ Theorem states that the conditional probability of si, given A, can be expressed as In the terminology of this section, P(si) is the prior probability of si and is modified to the posterior probability, P(si|A), given the sample information that event A has occurred Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall ))P(ss|P(A))P(ss|P(A))P(ss|P(A ))P(ss|P(A P(A) ))P(ss|P(A A)|P(s HH2211 iiii i Ch. 18-27 18.4
- 28. Bayes’ Theorem Example Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Stock Choice (Action) Percent Return (Events) Strong Economy (.7) Weak Economy (.3) Stock A 30 -10 Stock B 14 8 Consider the choice of Stock A vs. Stock B Expected Return: 18.0 12.2 Stock A has a higher EMV Ch. 18-28
- 29. Bayes’ Theorem Example Permits revising old probabilities based on new information Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall New Information Revised Probability Prior Probability (continued) Ch. 18-29
- 30. Bayes’ Theorem Example Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Additional Information: Economic forecast is strong economy When the economy was strong, the forecaster was correct 90% of the time. When the economy was weak, the forecaster was correct 70% of the time. Prior probabilities from stock choice example F1 = strong forecast F2 = weak forecast E1 = strong economy = 0.70 E2 = weak economy = 0.30 P(F1 | E1) = 0.90 P(F1 | E2) = 0.30 (continued) Ch. 18-30
- 31. Bayes’ Theorem Example Revised Probabilities (Bayes’ Theorem) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3.)E|F(P,9.)E|F(P 2111 3.)E(P,7.)E(P 21 875. )3)(.3(.)9)(.7(. )9)(.7(. )F(P )E|F(P)E(P )F|E(P 1 111 11 125. )F(P )E|F(P)E(P )F|E(P 1 212 12 (continued) Ch. 18-31
- 32. EMV with Revised Probabilities Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall EMV Stock A = 25.0 EMV Stock B = 11.25 Revised probabilities Pi Event Stock A xijPi Stock B xijPi .875 strong 30 26.25 14 12.25 .125 weak -10 -1.25 8 1.00 Σ = 25.0 Σ = 11.25 Maximum EMV Ch. 18-32
- 33. Expected Value of Sample Information, EVSI Suppose there are K possible actions and H states of nature, s1, s2, . . ., sH The decision-maker may obtain sample information. Let there be M possible sample results, A1, A2, . . . , AM The expected value of sample information is obtained as follows: Determine which action will be chosen if only the prior probabilities were used Determine the probabilities of obtaining each sample result: ))P(ss|P(A))P(ss|P(A))P(ss|P(A)P(A HHi22i11ii Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-33
- 34. Expected Value of Sample Information, EVSI For each possible sample result, Ai, find the difference, Vi, between the expected monetary value for the optimal action and that for the action chosen if only the prior probabilities are used. This is the value of the sample information, given that Ai was observed MM2211 )VP(A)VP(A)VP(AEVSI Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall (continued) Ch. 18-34
- 35. Expected Value of Perfect Information, EVPI Perfect information corresponds to knowledge of which state of nature will arise To determine the expected value of perfect information: Determine which action will be chosen if only the prior probabilities P(s1), P(s2), . . ., P(sH) are used For each possible state of nature, si, find the difference, Wi, between the payoff for the best choice of action, if it were known that state would arise, and the payoff for the action chosen if only prior probabilities are used This is the value of perfect information, when it is known that si will occur Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-35
- 36. Another way to view the expected value of perfect information Expected Value of Perfect Information EVPI = Expected monetary value under certainty – expected monetary value of the best alternative Expected Value of Perfect Information, EVPI HH2211 )WP(s)WP(s)WP(sEVPI Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall The expected value of perfect information (EVPI) is (continued) Ch. 18-36
- 37. Expected Value Under Certainty Expected value under certainty = expected value of the best decision, given perfect information Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Investment Choice (Action) Profit in $1,000’s (Events) Strong Economy (.3) Stable Economy (.5) Weak Economy (.2) Large factory Average factory Small factory 200 90 40 50 120 30 -120 -30 20 Example: Best decision given “Strong Economy” is “Large factory” 200 120 20 Value of best decision for each event: Ch. 18-37
- 38. Expected Value Under Certainty Now weight these outcomes with their probabilities to find the expected value: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Investment Choice (Action) Profit in $1,000’s (Events) Strong Economy (.3) Stable Economy (.5) Weak Economy (.2) Large factory Average factory Small factory 200 90 40 50 120 30 -120 -30 20 200 120 20 (continued) 200(.3)+120(.5)+20(.2) = 124 Expected value under certainty Ch. 18-38
- 39. Expected Value of Perfect Information Expected Value of Perfect Information (EVPI) EVPI = Expected profit under certainty – Expected monetary value of the best decision Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall so: EVPI = 124 – 81 = 43 Recall: Expected profit under certainty = 124 EMV is maximized by choosing “Average factory”, where EMV = 81 (EVPI is the maximum you would be willing to spend to obtain perfect information) Ch. 18-39
- 40. Utility Analysis Utility is the pleasure or satisfaction obtained from an action The utility of an outcome may not be the same for each individual Utility units are arbitrary Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-40 18.5
- 41. Utility Analysis Example: each incremental $1 of profit does not have the same value to every individual: A risk averse person, once reaching a goal, assigns less utility to each incremental $1 A risk seeker assigns more utility to each incremental $1 A risk neutral person assigns the same utility to each extra $1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall (continued) Ch. 18-41
- 42. Three Types of Utility Curves Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall $ $ $ Risk Aversion Risk Seeker Risk-Neutral Ch. 18-42
- 43. Maximizing Expected Utility Making decisions in terms of utility, not $ Translate $ outcomes into utility outcomes Calculate expected utilities for each action Choose the action to maximize expected utility Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-43
- 44. The Expected Utility Criterion Consider K possible actions, a1, a2, . . ., aK and H states of nature. Let Uij denote the utility corresponding to the ith action and jth state and Pj the probability of occurrence of the jth state of nature Then the expected utility, EU(ai), of the action ai is The expected utility criterion: choose the action to maximize expected utility If the decision-maker is indifferent to risk, the expected utility criterion and expected monetary value criterion are equivalent H 1j ijjiHHi22i11i UPUPUPUP)EU(a Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-44
- 45. Chapter Summary Described the payoff table and decision trees Defined opportunity loss (regret) Provided criteria for decision making If no probabilities are known: maximin, minimax regret When probabilities are known: expected monetary value Introduced expected profit under certainty and the value of perfect information Discussed decision making with sample information and Bayes’ theorem Addressed the concept of utility Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-45