1. MBA7025_10.ppt/Apr 7, 2015/Page 1 Georgia State University - Confidential MBA 7025 Statistical Business Analysis Decision Tree & Bayes’ Theorem Apr 7, 2015

2. MBA7025_10.ppt/Apr 7, 2015/Page 2 Georgia State University - Confidential Agenda Bayes Theorem Decision Tree Problems

3. MBA7025_10.ppt/Apr 7, 2015/Page 3 Georgia State University - Confidential Decision Trees A method of visually structuring the problem Effective for sequential decision problems Two types of branches –Decision nodes –Choice nodes –Terminal points Solving the tree involves pruning all but the best decisions Completed tree forms a decision rule

4. MBA7025_10.ppt/Apr 7, 2015/Page 4 Georgia State University - Confidential Decision Nodes Decision nodes are represented by Squares Each branch refers to an Alternative Action The expected return (ER) for the branch is –The payoff if it is a terminal node, or –The ER of the following node The ER of a decision node is the alternative with the maximum ER

5. MBA7025_10.ppt/Apr 7, 2015/Page 5 Georgia State University - Confidential Chance Nodes Chance nodes are represented by Circles Each branch refers to a State of Nature The expected return (ER) for the branch is –The payoff if it is a terminal node, or –The ER of the following node The ER of a chance node is the sum of the probability weighted ERs of the branches –ER =  P(S i ) * V i

6. MBA7025_10.ppt/Apr 7, 2015/Page 6 Georgia State University - Confidential Terminal Nodes Terminal nodes are optionally represented by Triangles The node refers to a payoff The value for the node is the payoff

7. MBA7025_10.ppt/Apr 7, 2015/Page 7 Georgia State University - Confidential Problem 1 Jenny Lind is a writer of romance novels. A movie company and a TV network both want exclusive rights to one of her more popular works. If she signs with the network, she will receive a single lump sum, but if she signs with the movie company the amount she will receive depends on the market response to her movie. Jenny Lind – Potential Payouts Movie company Small box office - $200,000 Medium box office - $1,000,000 Large box office - $3,000,000 TV Network Flat rate - $900,000 Questions: How can we represent this problem? What decision criterion should we use?

8. MBA7025_10.ppt/Apr 7, 2015/Page 8 Georgia State University - Confidential Jenny Lind – Payoff Table Decisions States of Nature Small Box OfficeMedium Box OfficeLarge Box Office Sign with Movie Company $200,000$1,000,000$3,000,000 Sign with TV Network$900,000

9. MBA7025_10.ppt/Apr 7, 2015/Page 9 Georgia State University - Confidential Jenny Lind – Decision Tree Small Box Office Medium Box Office Large Box Office Small Box Office Medium Box Office Large Box Office Sign with Movie Co. Sign with TV Network $200,000 $1,000,000 $3,000,000 $900,000

10. MBA7025_10.ppt/Apr 7, 2015/Page 10 Georgia State University - Confidential Problem 2 – Solving the Tree Start at terminal node at the end and work backward Using the ER calculation for decision nodes, prune branches (alternative actions) that are not the maximum ER When completed, the remaining branches will form the sequential decision rules for the problem

11. MBA7025_10.ppt/Apr 7, 2015/Page 11 Georgia State University - Confidential Jenny Lind – Decision Tree (Solved)

12. MBA7025_10.ppt/Apr 7, 2015/Page 12 Georgia State University - Confidential Decision Tree – Activation Test Source: Delta Airlines SkyMiles Program SkyMiles Enrollment Message A Returned within xx days Message B Returned within xx days Did not return within xx days Message C Did not return within xx days If Vc ¥ xx, send Message D Graduate to “SOW” Did not return within xx days If Vc < xx, no more messages Graduate to “SOW” If Vc ¥ xx, send Message D If Vc < xx, no more messages

13. MBA7025_10.ppt/Apr 7, 2015/Page 13 Georgia State University - Confidential Probability The Three Requirements of Probabilities: 1.All Probabilities must lie with the range of 0 to 1. 2.The sum of the individual probabilities equal to the probability of their union 3.The total probability of a complete set of outcomes must be equal to 1.

14. MBA7025_10.ppt/Apr 7, 2015/Page 14 Georgia State University - Confidential Direct Marketing Campaign Platform

15. MBA7025_10.ppt/Apr 7, 2015/Page 15 Georgia State University - Confidential Communication “Variables” Vehicles  = E-mail  = Kits  = Statement  = Telephone  = Direct Mail (USPS) Message / Offer (incentive) Hurdle (SOW) › trip x get y Next trip (Re-Activation) › Rate of trip triggers Points (double/flat?) Miles (front & back-end) Other Creative Execution Can test several executions tailored to clusters/segments Timing/Frequency Monthly (statements) Repeat/Follow-up Mailings

16. MBA7025_10.ppt/Apr 7, 2015/Page 16 Georgia State University - Confidential “Measuring Effectiveness: Lift/Gains Chart Percent of population targeted Percent of potential responders captured 100 0 90 45 Targeting Random mailing

17. MBA7025_10.ppt/Apr 7, 2015/Page 17 Georgia State University - Confidential Example Direct Mail Optimization Source: InterContinental Hotels Group Priority Club Rewards Program Using multivariate model we are able to maximize profit while minimizing costs In comparison to methodology used last year model savings = $XXX –Savings attributable to reduced mailing to achieve last years result (variable cost savings). Other benefits - Customer Behavior, Planning Tool

18. MBA7025_10.ppt/Apr 7, 2015/Page 18 Georgia State University - Confidential Agenda Decision Tree Bayes Theorem Problems

19. MBA7025_10.ppt/Apr 7, 2015/Page 19 Georgia State University - Confidential Bayes' Theorem Bayes' Theorem is used to revise the probability of a particular event happening based on the fact that some other event had already happened. Probabilities Involved P(Event) Prior probability of this particular situation P(Prediction | Event) Predictive power (Likelihood) of the information source P(Prediction  Event) Joint probabilities where both Prediction and Event occur P(Prediction) Marginal probability that this prediction is made P(Event | Prediction) Posterior probability of Event given Prediction

20. MBA7025_10.ppt/Apr 7, 2015/Page 20 Georgia State University - Confidential Bayes’ Theorem Bayes's Theorem begins with a statement of knowledge prior to performing the experiment. Usually this prior is in the form of a probability density. It can be based on physics, on the results of other experiments, on expert opinion, or any other source of relevant information. Now, it is desirable to improve this state of knowledge, and an experiment is designed and executed to do this. Bayes's Theorem is the mechanism used to update the state of knowledge to provide a posterior distribution. The mechanics of Bayes's Theorem can sometimes be overwhelming, but the underlying idea is very straightforward: Both the prior (often a prediction) and the experimental results have a joint distribution, since they are both different views of reality.joint distribution

21. MBA7025_10.ppt/Apr 7, 2015/Page 21 Georgia State University - Confidential Bayes’ Theorem Let the experiment be A and the prediction be B. Both have occurred, AB. The probability of both A and B together is P(AB). The law of conditional probability says that this probability can be found as the product of the conditional probability of one, given the other, times the probability of the other. That is P(A|B) ´ P(B) = P(AB) = P(B|A) ´ P(A) if both P(A) and P(B) are non zero. Simple algebra shows that: P(B|A) = P(A|B) ´ P(B) / P(A) equation 1 This is Bayes's Theorem. In words this says that the posterior probability of B (the updated prediction) is the product of the conditional probability of the experiment, given the influence of the parameters being investigated, times the prior probability of those parameters. (Division by the total probability of A assures that the resulting quotient falls on the [0, 1] interval, as all probabilities must.)

22. MBA7025_10.ppt/Apr 7, 2015/Page 22 Georgia State University - Confidential Bayes’ Theorem

23. MBA7025_10.ppt/Apr 7, 2015/Page 23 Georgia State University - Confidential Conditional Probability

24. MBA7025_10.ppt/Apr 7, 2015/Page 24 Georgia State University - Confidential Bayes' Theorem

25. MBA7025_10.ppt/Apr 7, 2015/Page 25 Georgia State University - Confidential Probability Information Prior Probabilities –Initial beliefs or knowledge about an event (frequently subjective probabilities) Likelihoods –Conditional probabilities that summarize the known performance characteristics of events (frequently objective, based on relative frequencies)

26. MBA7025_10.ppt/Apr 7, 2015/Page 26 Georgia State University - Confidential Circumstances for using Bayes’ Theorem You have the opportunity, usually at a price, to get additional information before you commit to a choice You have likelihood information that describes how well you should expect that source of information to perform You wish to revise your prior probabilities

27. MBA7025_10.ppt/Apr 7, 2015/Page 27 Georgia State University - Confidential Problem A company is planning to market a new product. The company’s marketing vice-president is particularly concerned about the product’s superiority over the closest competitive product, which is sold by another company. The marketing vice-president assessed the probability of the new product’s superiority to be 0.7. This executive then ordered a market survey to determine the products superiority over the competition. The results of the survey indicated that the product was superior to its competitor. Assume the market survey has the following reliability: –If the product is really superior, the probability that the survey will indicate “superior” is 0.8. –If the product is really worse than the competitor, the probability that the survey will indicate “superior” is 0.3. After completion of the market survey, what should the vice-president’s revised probability assignment to the event “new product is superior to its competitors”?

28. MBA7025_10.ppt/Apr 7, 2015/Page 28 Georgia State University - Confidential Joint Probability Table P(Ai)P(B|Ai)P(Ai)* P(B|Ai)Revised Probability P(Ai|B) A1 Probability product is superior 0.70.80.560.56/0.65 = 0.86 A2 Probability product is not superior 0.3 0.090.09/0.65 = 0.14 1.0P(B) =0.65

29. MBA7025_10.ppt/Apr 7, 2015/Page 29 Georgia State University - Confidential Agenda Decision Tree Bayes Theorem Problems

30. MBA7025_10.ppt/Apr 7, 2015/Page 30 Georgia State University - Confidential What kinds of problems? Alternatives known States of Nature and their probabilities are known. Payoffs computable under different possible scenarios

31. MBA7025_10.ppt/Apr 7, 2015/Page 31 Georgia State University - Confidential Basic Terms Decision Alternatives States of Nature (eg. Condition of economy) Payoffs ($ outcome of a choice assuming a state of nature) Criteria (eg. Expected Value) Z

32. MBA7025_10.ppt/Apr 7, 2015/Page 32 Georgia State University - Confidential Example Problem 1 - Expected Value & Decision Tree

33. MBA7025_10.ppt/Apr 7, 2015/Page 33 Georgia State University - Confidential Expected Value

34. MBA7025_10.ppt/Apr 7, 2015/Page 34 Georgia State University - Confidential Decision Tree

35. MBA7025_10.ppt/Apr 7, 2015/Page 35 Georgia State University - Confidential Example Problem 2 - Sequential Decisions Would you hire a consultant (or a psychic) to get more info about states of nature? How would additional info cause you to revise your probabilities of states of nature occurring? Draw a new tree depicting the complete problem. Consultant’s Track Record Z

36. MBA7025_10.ppt/Apr 7, 2015/Page 36 Georgia State University - Confidential Example Problem 2 - Sequential Decisions (Ans) Open MBA7020Joint_Probabilities_Table.xls 1.First thing you want to do is get the information (Track Record) from the Consultant in order to make a decision. 2.This track record can be converted to look like this: P(F/S1) = 0.2P(U/S1) = 0.8 P(F/S2) = 0.6P(U/S2) = 0.4 P(F/S3) = 0.7P(U/S3) = 0.3 F= FavorableU=Unfavorable 3.Next, you take this information and apply the prior probabilities to get the Joint Probability Table/Bayles Theorum Z

37. MBA7025_10.ppt/Apr 7, 2015/Page 37 Georgia State University - Confidential Example Problem 2 - Sequential Decisions (Ans) Open MBA7020Joint_Probabilities_Table.xls 4.Next step is to create the Posterior Probabilities (You will need this information to compute your Expected Values) P(S1/F) = 0.06/0.49 = 0.122 P(S2/F) = 0.36/0.49 = 0.735 P(S3/F) = 0.07/0.49 = 0.143 P(S1/U) = 0.24/0.51 = 0.47 P(S2/U) = 0.24/0.51 = 0.47 P(S3/U) = 0.03/0.51 = 0.06 5.Solve the decision tree using the posterior probabilities just computed. Z