Decision Theory at its Best

Transcription

1 Decision Theory at its Best I am very excited about today s class. Today s Fare: Bayesian Revision

2 Bayesian Decision Making The Reverend Bayes shows us how to do it with posterior distributions. See priors and likelihoods come together. Decision making with experimentation You will not want to miss this! See it happen in front of your very own eyes.

3 Thomas Bayes Mathematician who first used probability inductively and established a mathematical basis for probability inference (a means of calculating, from the number of times an event has not occurred, the probability that it will occur in future trials). He set down his findings on probability in "Essay Towards Solving a Problem in the Doctrine of Chances" (1763), published posthumously in the Philosophical Transactions of the Royal Society of London. Bayes' contributions are immortalized by naming a fundamental proposition in probability, called Bayes Rule, after him. Born: 1702 in London, England Died: 17 April 1761 in Tunbridge Wells, Kent, England

4 Let s Review Bayes Theorem A test for a particular disease is positive 99 % of the time if the disease is present. This same test is negative 95% of the time if the disease is not present. Five persons in every thousand have this disease. How good is the test when administered to the population at random?

5 We are still reviewing Bayes Theorem Define events: A = the event, the disease is present B = the event, test is positive Summarize what is known: P(A) =.005 (prior) P(B A) =.99; P(B c A c ) =.95; (likelihoods) What is desired: P(A B) =?; P(A c B c ) =? (posterior)

6 Yes, still reviewing The infamous Bayes Theorem: PAB ( ) PA ( B) PB ( APA ) ( ) = = c c P( B) P( B A) P( A) + P( B A ) P( A ) A B

7 The Calculations PB ( APA ) ( ) PAB ( ) = c c PB ( APA ) ( ) + PB ( A) PA ( ) (.99)(.005) = =.0905 (.99)(.005) +.05(.995) c c c c c PB ( A) PA ( ) PA ( B) = c c c c P( B A ) P( A ) + P( B A) P( A) (.95)(.995) = = (.95)(.995) +.01(.005)

8 The Whole Thing Am I not sick? disease (.005) test positive (.99) test negative (.01) no disease (.995) test positive (.05) PA ( B) PAB ( ) = = =.0905 PB ( ) c c c c PA ( B) PA ( B) = = = c PB ( ) test negative (.95)

9 The Three Distributions Prior - P(A) theoretical, empirical, subjective Likelihood - P(B A) and P(B A c ) conditional probabilities Posterior P(A B) integrates prior with likelihood using Bayes theorem

10 Generalize Bayes Theorem PB ( A) ( ) ( ) j PAj P Aj B = n ; j = 1, 2,..., n PB ( Ai) PA ( i) i= 1 P(A j ) = prior distribution B = event, sampling (new) information P(B A i ) = likelihood distribution P(A j B) = posterior distribution

11 Why does it work? A 1 B A 2 A 3

12 Why does it work? PB ( A) = PBA ( ) PA ( ) P( B A ) = P( B A ) P( A ) A 1 B A 2 A 3 P( B A ) = P( B A ) P( A ) 3 3 3

13 The Decision Table State of Nature Prior Likelihood Product Posterior A 1 P(A 1 ) P(B A 1 ) P(B A 1 ) P(A 1 ) P(A 1 B)= P(B A 1 ) P(A 1 )/P(B) A 2 P(A 2 ) P(B A 2 ) P(B A 2 ) P(A 2 ) P(A 2 B)= P(B A 2 ) P(A 1 )/P(B)... A n P(A n ) P(B A n ) P(B A n ) P(A n ) P(A n B)= P(B A n ) P(A n )/P(B) SUM 1.0 P(B) 1.0 n PB ( ) = PBA ( ) PA ( ) i= 1 i i

14 The Inevitable Urn Problem 5 red 5 green 2 red 8 green 8 red 2 green Urn I Urn II Urn III Select an urn and draw a ball at random. Prior: P(Urn I) = P(Urn II) = P(Urn III) = 1/3 Let B = event, draw a red ball Likelihoods: P(Red Urn I) =.5 P(Red Urn II) =.2 P(Red Urn III) =.8

15 The Urns & the Decision Table Urn Prior Likelihood Joint Prob Posterior 1 1/3 1/2 1/6 (1/6)/(1/2)=1/3 2 1/3 1/5 1/15 (1/15)/(1/2)=2/15 3 1/3 4/5 4/15 (4/15)/(1/2)=8/ P(Red)=1/2 1.0

16 Returning to our investment decision... Investment decision: $1,000 to invest. probability solid alternatives growth stagnation inflation bonds stocks credit union expected value E(bonds) =.5 (12) +.3 (8) +.2 (4) = 9.2 E(stocks) =.5 (15) +.3 (3) +.2 (-2) = 8.0 E(CU) =.5 (7) +.3 (7) +.2 (7) = 7.0

17 The Likelihood Distribution Can purchase an economic forecast having the following historical performance: When observed Forecast predicted Growth Stagnation Inflation Growth Stagnation Inflation Pr(Forecast Stagnation Stagnation) Pr(Forecast Stagnation Inflation)

18 Revision - The Posterior Forecast says GROWTH: Prior Likelihood Joint Posterior Growth Stagnation Inflation P(forecast growth) = Forecast says STAGNATION: Prior Likelihood Joint Posterior Growth Stagnation Inflation P(forecast Stagnation) =

19 More Bayesian Revision Forecast says INFLATION: Prior Likelihood Joint Posterior Growth Stagnation Inflation P(forecast Inflation) = I wholeheartedly endorse this Bayesian thing.

20 Forecast says GROWTH: Prior Likelihood Joint Posterior Growth Stagnation Inflation P(forecast growth) =.38 Forecast says STAGNATION: Prior Likelihood Joint Posterior Growth Stagnation Inflation P(forecast Stagnation) =.35 Forecast says INFLATION: Prior Likelihood Joint Posterior Growth Stagnation Inflation (forecast Inflation) =

21 The Pre-Posterior Analysis- 1 If GROWTH is forecasted: E(bonds) =.921(12) (8) + 0 (4) = E(stocks) =.921 (15) (3) + 0 (-2) = E(credit union) = 7.0 Posterior prob solid alternatives growth stagnation inflation bonds stocks credit union Can I watch?

22 The Pre-Posterior Analysis - 2 If STAGNATION is forecasted: E(bonds) =.143(12) (8) (4) = 7.89 E(stocks) =.143 (15) (3) (-2) = 3.86 E(credit union) = 7.0 Posterior prob solid alternatives growth stagnation inflation bonds stocks credit union Can I watch?

23 The Pre-Posterior Analysis -3 If INFLATION is forecasted: E(bonds) =.370 (12) +.111(8) (4) = 7.40 E(stocks) =.370 (15) (3) (-2) = 4.85 E(credit union) = 7.0 Posterior prob solid alternatives growth stagnation inflation bonds stocks credit union Can I watch?

24 The Pre-Posterior Analysis - 4 If GROWTH is forecasted: E(bonds) =.921(12) (8) + 0 (4) = E(stocks) =.921 (15) (3) + 0 (-2) = E(credit union) = 7.0 If STAGNATION is forecasted: E(bonds) =.143(12) (8) (4) = 7.89 E(stocks) =.143 (15) (3) (-2) = 3.86 E(credit union) = 7.0 If INFLATION is forecasted: E(bonds) =.370 (12) +.111(8) (4) = 7.40 E(stocks) =.370 (15) (3) (-2) = 4.85 E(credit union) = 7.0 EVSI =.38 (14.05) +.35 (7.89) +.27 (7.40) = =.9 or.009 x $1,000 = $ 9.00

25 Can we now do some more Bayesian revisions? Please!

26 The Sickly Mining Company The Red Shield Insurance Company is considering offering a group health insurance plan to the Sickly Mining Company. The Sickly Mining Company operates a half dozen coal mines in the mountains of West Virginia. The profit that would be realized by Red Shield depends upon the number of employees that sign up for the insurance. Red Shield can at a cost of $2500 survey the Sickly work force concerning employee interest in the health plan.

27 Some Data Current employee participation in Red Shield Health Plans number of companies High Participation (> 80%) 30 Medium Participation (50% to 80%) 50 Low Participation (< 50%) 20 Historical Survey Results: Survey Prediction Actual Participation High medium Low High Medium Low 1 2 7

28 The New Health Plan I need to get me some of that. Prior: Participation level: High Moderate Low EMV Introduce plan 60,000 20,000-70,000 14,000 Do not intro plan EVPI =.3 (60,000) +.5 (20,000) +.2 (0) - 14,000 = $14,000

29 The New Health Plan I need to get me some of that. Prior Demand High Moderate Low EMV Introduce 60,000 20,000-70,000 14,000 Do not intro EVPI =.3 (60,000) +.5 (20,000) +.2 (0) - 14,000 = $14,000 State Likelihood Joint Posterior of nature Prior High Mod Low High Mod Low High Mod Low High Mod Low

30 Why didn t I make the optimal Bayesian decision? I said no to the survey! The New Health Plan E(profit High) =.72 (60,000) +.20 (20,000) -.08 (70,000) = 41,600 E(profit Mod) =.12 (60,000) +.80 (20,000) -.08 (70,000) = 17,600 E(profit Low) =.24 (60,000) +.20 (20,000) -.56 (70,000) = -20,800 EVSI =.25 (41,600) +.50 (17,600) -.25 (0) - 14,000 = $ 5,200 Optimal Decision: Red Shield should pay for the survey. If the survey forecasts high or moderate demand then introduce the new plan; otherwise do not introduce it.

31 The Application of Basic Business Principles to Day-to-Day Living See? I told you it takes money to make money. The Hideout