Chapter 15. Bayesian Statistics and Decision Analysis. COMPLETE BUSINESS STATISTICS by AMIR D. ACZEL & JAYAVEL SOUNDERPANDIAN

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1 COMPLETE BUSINESS STATISTICS by AMIR D. ACZEL & JAYAVEL SOUNDERPANDIAN 6 th editio Chater 15 Bayesia Statistics Decisio Aalysis Bayesia Statistics Decisio Aalysis LEARNING OBJECTIVES 15-4 Usig Statistics Bayes Theorem Discrete Probability Models Bayes Theorem Cotiuous Probability Distributios The Evaluatio of Subjective Decisio Aalysis: A Overview Decisio Trees Hlig Additioal Iformatio Usig Bayes Theorem The Value of Iformatio Usig the Comuter After studyig this chater you should be able to: Aly Bayes theorem to revise oulatio arameters Solve sequetial decisio roblems usig decisio trees Coduct decisio aalysis for cases out robability data Coduct decisio aalysis for cases robability data

2 LEARNING OBJECTIVES (2) Bayesia Classical Statistics After studyig this chater you should be able to: Evaluate the eected value of erfect iformatio Evaluate the eected value of samle iformatio Use utility fuctios to model the risk attitudes of decisio makers Solve decisio aalysis roblems usig sreadsheet temlates Data Data Prior Iformatio Classical Iferece Bayesia Iferece Statistical Coclusio Statistical Coclusio Bayesia statistical aalysis icororates a rior robability distributio likelihoods of observed data to determie a osterior robability distributio of evets. Bayes Theorem: Eamle Bayes Theorem Discrete Probability Models _ Eamle (Cotiued) 15-8 A medical test for a rare disease (affectig % of of the oulatio [ I) ]) ]) is is imerfect: Whe admiistered to to a a ill ill erso, the the test will idicate so so robability 0.92 [[ PZI ( ). 92 PZI ( ). 08] ] The evet ( ZI) is is a false egative Whe admiistered to to a erso who is is ot ill, ill, the the test will erroeously give a ositive result (false ositive) robability 0.04 [[ PZI ( ) 004. PZI ( ) 096. ]] The evet ( ZI) is is a false ositive... Alyig Bayes Theorem I) PI ( ) PZI ( ) 092. PZI ( ) 004. P ( I I Z ) I Z) Z) I I Z) I I Z) + I I Z ) Z I) I) Z I) I) + Z I ) I ) (. 92)( ) (. 92)( ) + ( 0. 04)(. 999)

3 Eamle 2-10: 2 Decisio Tree Bayes Theorem Discrete Probability Models Prior Coditioal Joit PZI ( ) 092. Z I I) ( )( 0. 92) The likelihood fuctio is the set of coditioal robabilities θ) for give data, cosiderig a fuctio of a ukow oulatio arameter, θ. ) I I ) Z I) 0.08 Z I ) 0.04 Z I ) 0.96 ZI I) ( )( 008. ) Z I I ) (0.999)(0.04) Z I I ) (0.999)(0.96) Bayes theorem for a discrete rom variable: P ( θ) θ) θ ) P ( θ ) θ ) i where θ is a ukow oulatio arameter to be estimated from the data. The summatio i the deomiator is over all ossible values of the arameter of iterest, θ i, sts for the observed data set. i i Eamle 15-1: 1: Prior Distributio Likelihoods of 4 Successes i Trials Eamle 15-1: 1: Prior, Likelihoods, Posterior Prior Distributio S S) Likelihood Biomial Biomial X ) ) Biomial Biomial X ) ) Biomial Biomial X ) ) Biomial Biomial X ) ) Biomial Biomial X ) ) Biomial Biomial X ) ) Prior Posterior Distributio Likelihood Distributio S S) S) S) S) S ) % Credible Set

4 Eamle 15-1: 1: Prior Posterior Distributios Eamle 15-1: 1: A Secod Samlig 3 Successes i Trials S) P osterior D is tributio of Marke t S hare S P (S) Prior Distributio of Share S Prior Distributio S S) Likelihood Biomial Biomial X ) ) Biomial Biomial X ) ) Biomial Biomial X ) ) Biomial Biomial X ) ) Biomial Biomial X ) ) Biomial Biomial X ) ) Eamle 15-1: 1: Icororatig a Secod Samle Eamle 15-1: 1: Usig the Temlate 15- Prior Posterior Distributio Likelihood Distributio S S) S) S) S) S ) % Credible Set Alicatio of of Bayes Theorem usig the Temlate. The osterior robabilities are calculated usig a formula based o o Bayes Theorem for discrete rom variables.

5 Eamle 15-1: 1: Usig the Temlate (Cotiued) Bayes Theorem Cotiuous Probability Distributios We defie f(θ) as as the therior robability desity of of the the arameter θ. θ. We defie f( θ) as as the the coditioal desity of of the the data,, give the the value of of θ.. This is is the the likelihood fuctio. Dislay of the Prior Posterior robabilities. Bayes' theorem for cotiuous distributios: f ( θ ) f ( θ) f ( θ) f ( θ) f ( θ) f ( θ) f ( θ) dθ Total area uder f (,θ) The Normal Probability Model The Normal Probability Model: Eamle Normal oulatio ukow mea µ kow stard deviatio σ Poulatio mea is is a rom variable ormal (rior) distributio mea M M stard deviatio σ. σ. Draw samle of of size : : The osterior mea variace of the ormal oulatio of the oulatio mea, µ : σ M M σ 2 1 M σ σ + σ σ σ M 15 σ 8 10 M s M M σ σ 2 1 M σ σ σ σ σ M σ M σ σ % Credible Set: M ± 196. σ ± ( 196. ) [ , ]

6 Eamle 15-2 Eamle Usig the Temlate Desity Posterior Distributio Likelihood Prior Distributio µ Eamle Usig the Temlate (Cotiued) The Evaluatio of Subjective Based o ormal distributio 95% of ormal distributio is i 2 stard deviatios of the mea -1 < < 31).95 µ 15, σ 8 68% of ormal distributio is i 1 stard deviatio of the mea 7 < < 23).68 µ 15, σ 8

7 Decisio Aalysis : Decisio Tree: New-Product Itroductio Elemets of a decisio aalysis Actios Aythig the the decisio-maker ca ca do do at at ay ay time time Chace occurreces Possible outcomes (samle (samle sace) sace) associated chace occurreces Fial outcomes Payoff, reward, reward, or or loss loss associated actio actio Additioal iformatio Allows decisio-maker to to reevaluate robabilities ossible ossible rewards rewards losses losses Decisio Course of of actio actio to to take take i i each each ossible ossible situatio Decisio Do ot market Chace Occurrece Product successful (P 0.75) Product usuccessful (P 0.25) Fial Outcome $100,000 -$,000 $0 15-6: Payoff Table Eected Values of Decisios: New-Product Itroductio Solutio to the New-Product Itroductio Decisio Tree Product is is Actio Successful Not Successful the the roduct $100,000 -$,000 Do Do ot market the the roduct $0 $0 $0 $0 Cliig the Nootimal Decisio Braches Eected Payoff Payoff $70,000 $70,000 Product successful (P0.75) $100,000 The eected value of X, deoted E( X ): E( X) ) all EOutcome ( ) (100, 000)( 0. 75) + (, 000)( 0. 25) ,000 Nootimal decisio decisio brach brach is is clied clied Do ot market Eected Payoff Payoff $0 $0 Product usuccessful (P0.25) -$,000 $0

8 New-Product Itroductio: Eteded-Possibilities New-Product Itroductio: Eteded-Possibilities Decisio Tree Outcome Payoff Probability ) Etremely successful $150,000 15,000 Very successful ,000 Successful 100, ,000 Somewhat successful 80,000 8,000 Barely successful 40,000 4,000 Break eve 0 0 Usuccessful -, Disastrous -50, ,500 Eected Payoff: $77,500 Nootimal decisio decisio brach brach is is clied clied Decisio Do ot market Eected Payoff Payoff $77,500 $77,500 Chace Occurrece $100,000 $80,000 $40,000 $0 -$,000 -$50,000 $0 Payoff $150,000 $1, Eamle 15-3: Decisio Tree Eamle 15-3: Solutio Lease Not Lease Pr 0.5 Not Promote Promote $700,000 P r 0.4 $680,000 Pr 0.6 $740,000 Pr 0.3 $800,000 Pr 5 $900,000 Pr 0.05 $1,000,000 Pr 0.9 $750,000 Pr $780,000 Lease Not Lease Eected ayoff: 0.5* *7000 $783,000 Eected ayoff: $753,000 Pr0.5 Not Promote Promote Eected ayoff: $425,000 Eected ayoff: $700,000 Eected ayoff: $7,000 Pr 0.4 Pr 0.6 Pr 0.3 Pr 5 Pr 0.05 Pr 0.9 Pr $700,000 $680,000 $740,000 $800,000 $900,000 $1,000,000 $750,000 $780,000

9 Hlig Additioal Iformatio Usig Bayes Theorem Alyig Bayes Theorem Test Not test Test idicates success Test idicates failure New-Product Decisio Tree Tree Testig Testig Do ot market Do ot market Do ot market Successful Failure Successful Failure Successful Failure Pr0.75 Pr0.25 Payoff $95,000 -$25,000 -$5,000 $95,000 -$25,000 -$5,000 $100,000 -$,000 0 S)0.75 IS S)0.9 IF S) F)0.75 IS F)5 IF S)0.85 IS)IS S)S)+IS F)F)(0.9)(0.75)+(5)(0.25) IF)IF S)S)+IF F)F)()(0.75)+(0.85)(0.25) IS S)S) S IS) IS S)S) + IS F)F) ( 09. )( 075. ) ( 09. )( 075. ) + ( 015. )( 025. ) F IS) 1 S IS) IF S)S) S IF) IF S)S) + IF F)F) ( 01. )( 075. ) ( 01. )( 075. ) + ( 085. )( 025. ) F IF) 1 S IF) Eected Payoffs Solutio Eamle 15-4: Payoffs Test $70,000 Not test $ IS) IF) $86,866 $86,866 S IS) F IS) Do ot market $6,308 Do ot market $70,000 Do ot market $6,308 $70,000 S IF) F IF) S)0.75 F)0.25 $95,000 -$25,000 -$5,000 $95,000 -$25,000 -$5,000 $100,000 -$,000 0 Payoff Prior Prior Iformatio Reliability of of Cosultig Firm Firm Level Level of of Future Future Ecoomic State State of of Cosultats Coclusio Profit Profit Activity Probability Ecoomy High High Medium Low Low $3 $3 millio millio Low Low Low Low $6 $6 millio millio Medium Medium $12 $12 millio millio High High High High Cosultats say say Low Low Evet Evet Prior PriorCoditioal Joit Joit Posterior Low Low Medium High High Cosultats say say Low ) Low )

10 Eamle 15-4: Joit Coditioal Eamle 15-4: Decisio Tree Cosultats say say Medium Evet Evet Prior PriorCoditioal Joit Joit Posterior Low Low Medium High High Cosultats say say Medium ) Cosultats say say High High Evet Evet Prior PriorCoditioal Joit Joit Posterior Low Low Medium High High Cosultats say say High ) Alterative Ivestmet Profit Profit Probability $4 $4 millio millio $7 $7 millio millio Cosultig fee: fee: $1 $1 millio millio Alterative $7 millio $4 millio $12 millio Ivest $6 millio Do ot hire cosultats $3 millio Alterative $6 millio $3 millio $11 millio Ivest $5 millio $2 millio Hire cosultats 5.5 H 7.2 L 4.5 H L 4.5 H L H L M M M $6 millio H Alterative $3 millio L $11 millio Ivest $5 millio $2 millio Alterative $6 millio $3 millio $11millio Ivest M M $5 millio $2 millio Margial Attitudes toward Risk is is a a measure of of the the total total worth worth of of a a articular outcome. It It reflects reflects the the decisio maker s maker s attitude attitude toward toward a a collectio of of factors factors such such as as rofit, rofit, loss, loss, risk. risk. Risk Averse Risk Taker Additioal Additioal { Risk Neutral Mied } Additioal $1000 } Additioal $1000

11 Eamle 15-5: 5: Assessig The Value of Iformatio Possible Iitial Iitial Idifferece Returs $1,500 $1, ,300 4,300 (1500)(0.8)+(56000)(0.2) ,000 22,000 (1500)(0.3)+(56000)(0.7) ,000 31,000 (1500)(0.2)+(56000)(0.8) ,000 56, The The eected value value of of erfect erfect iformatio (EVPI): EVPI EVPI The The eected moetary value value of of the the decisio situatio whe whe erfect erfect iformatio is is available mius mius the the eected value value of of the the decisio situatio whe whe o o additioal iformatio is is available. Eected Net Gai Eected Net Gai from Samlig Eected Net Gai from Samlig Ma ma Samle Size Eamle 15-6: The Decisio Tree Eamle 15-6: Value of Additioal Iformatio Airlie Fare $0 Fare $300 Fare Cometitor s Fare Cometitor:$0 Pr 0.6 Cometitor:$300 Pr 0.4 Cometitor:$0 Pr 0.6 Cometitor:$300 Pr 0.4 Payoff $8 millio $9 millio $4 millio $10 millio If If o additioal iformatio is is available, the best strategy is is to set the fare at $0 E(Payoff 0) (.6)(8)+(.4)(9) $8.4 millio E(Payoff 300) (.6)(4)+(.4)(10) $6.4 millio With further iformatio, the eected ayoff could be: E(Payoff Iformatio) (.6)(8)+(.4)(10)$8.8 millio EVPI $.4 millio