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15-1 COMPLETE BUSINESS STATISTICS by AMIR D. ACZEL & JAYAVEL SOUNDERPANDIAN 6 th edition (SIE)

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15-2 Chapter 15 Bayesian Statistics and Decision Analysis

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15-3 Using Statistics Bayes Theorem and Discrete Probability Models Bayes Theorem and Continuous Probability Distributions The Evaluation of Subjective Probabilities Decision Analysis: An Overview Decision Trees Handling Additional Information Using Bayes Theorem Utility The Value of Information Using the Computer Bayesian Statistics and Decision Analysis 15

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15-4 Apply Bayes theorem to revise population parameters Solve sequential decision problems using decision trees Conduct decision analysis for cases without probability data Conduct decision analysis for cases with probability data LEARNING OBJECTIVES 15 After studying this chapter you should be able to:

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15-5 Evaluate the expected value of perfect information Evaluate the expected value of sample information Use utility functions to model the risk attitudes of decision makers Solve decision analysis problems using spreadsheet templates LEARNING OBJECTIVES (2) 15 After studying this chapter you should be able to:

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15-6 Classical Inference Classical Inference Data Statistical Conclusion Statistical Conclusion Bayesian Inference Bayesian Inference Data Prior Information Prior Information Statistical Conclusion Statistical Conclusion Bayesian statistical analysis incorporates a prior probability distribution and likelihoods of observed data to determine a posterior probability distribution of events. Bayesian and Classical Statistics

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15-7 A medical test for a rare disease (affecting 0.1% of the population [ ]) is imperfect: When administered to an ill person, the test will indicate so with probability 0.92 [ ] The event is a false negative When administered to a person who is not ill, the test will erroneously give a positive result (false positive) with probability 0.04 [ ] The event is a false positive.. A medical test for a rare disease (affecting 0.1% of the population [ ]) is imperfect: When administered to an ill person, the test will indicate so with probability 0.92 [ ] The event is a false negative When administered to a person who is not ill, the test will erroneously give a positive result (false positive) with probability 0.04 [ ] The event is a false positive.. Bayes Theorem: Example 2-10

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15-8 Applying Bayes Theorem 15-2 Bayes Theorem and Discrete Probability Models _ Example 2-10 (Continued)

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15-9 Prior Probabilities Prior Probabilities Conditional Probabilities Conditional Probabilities Joint Probabilities Joint Probabilities Example 2-10: Decision Tree

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15-10 Bayes theorem for a discrete random variable: where is an unknown population parameter to be estimated from the data. The summation in the denominator is over all possible values of the parameter of interest, i, and x stands for the observed data set. Bayes theorem for a discrete random variable: where is an unknown population parameter to be estimated from the data. The summation in the denominator is over all possible values of the parameter of interest, i, and x stands for the observed data set. The likelihood function is the set of conditional probabilities P(x| ) for given data x, considering a function of an unknown population parameter, Bayes Theorem and Discrete Probability Models

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15-11PriorDistribution SP(S) PriorDistribution SP(S) Likelihood Binomial with n = 20 and p = x P( X = x) Binomial with n = 20 and p = x P( X = x) Binomial with n = 20 and p = x P( X = x) Binomial with n = 20 and p = x P( X = x) Binomial with n = 20 and p = x P( X = x) Binomial with n = 20 and p = x P( X = x) Likelihood Binomial with n = 20 and p = x P( X = x) Binomial with n = 20 and p = x P( X = x) Binomial with n = 20 and p = x P( X = x) Binomial with n = 20 and p = x P( X = x) Binomial with n = 20 and p = x P( X = x) Binomial with n = 20 and p = x P( X = x) Example 15-1: Prior Distribution and Likelihoods of 4 Successes in 20 Trials

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15-12 Prior Posterior Distribution Likelihood Distribution SP(S)P(x|S) P(S)P(x|S) P(S|x) Prior Posterior Distribution Likelihood Distribution SP(S)P(x|S) P(S)P(x|S) P(S|x) % Credible Set Example 15-1: Prior Probabilities, Likelihoods, and Posterior Probabilities

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S P ( S ) Posterior Distribution of Market Share S P ( S ) Prior Distribution of Market Share Example 15-1: Prior and Posterior Distributions

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15-14 Prior Distribution SP(S) Prior Distribution SP(S) Likelihood Binomial with n = 16 and p = x P( X = x) Binomial with n = 16 and p = x P( X = x) Binomial with n = 16 and p = x P( X = x) Binomial with n = 16 and p = x P( X = x) Binomial with n = 16 and p = x P( X = x) Binomial with n = 16 and p = x P( X = x) Likelihood Binomial with n = 16 and p = x P( X = x) Binomial with n = 16 and p = x P( X = x) Binomial with n = 16 and p = x P( X = x) Binomial with n = 16 and p = x P( X = x) Binomial with n = 16 and p = x P( X = x) Binomial with n = 16 and p = x P( X = x) Example 15-1: A Second Sampling with 3 Successes in 16 Trials

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15-15 Prior Posterior Distribution Likelihood Distribution S P(S) P(x|S) P(S)P(x|S) P(S|x) Prior Posterior Distribution Likelihood Distribution S P(S) P(x|S) P(S)P(x|S) P(S|x) % Credible Set Example 15-1: Incorporating a Second Sample

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15-16 Application of Bayes Theorem using the Template. The posterior probabilities are calculated using a formula based on Bayes Theorem for discrete random variables. Application of Bayes Theorem using the Template. The posterior probabilities are calculated using a formula based on Bayes Theorem for discrete random variables. Example 15-1: Using the Template

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15-17 Example 15-1: Using the Template (Continued) Display of the Prior and Posterior probabilities.

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15-18 We define f( ) as the prior probability density of the parameter. We define f(x| ) as the conditional density of the data x, given the value of. This is the likelihood function Bayes Theorem and Continuous Probability Distributions

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15-19 Normal population with unknown mean and known standard deviation Population mean is a random variable with normal (prior) distribution and mean M and standard deviation. Draw sample of size n: Normal population with unknown mean and known standard deviation Population mean is a random variable with normal (prior) distribution and mean M and standard deviation. Draw sample of size n: The Normal Probability Model

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15-20 The Normal Probability Model: Example 15-2

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15-21 Likelihood Posterior Distribution Posterior Distribution Prior Distribution Prior Distribution 15 Density Example 15-2

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15-22 Example 15-2 Using the Template

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15-23 Example 15-2 Using the Template (Continued)

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15-24 Based on normal distribution Based on normal distribution 95% of normal distribution is within 2 standard deviations of the mean P(-1 < x < 31) =.95 = 15, = 8 68% of normal distribution is within 1 standard deviation of the mean P(7 < x < 23) =.68 = 15, = 8 Based on normal distribution Based on normal distribution 95% of normal distribution is within 2 standard deviations of the mean P(-1 < x < 31) =.95 = 15, = 8 68% of normal distribution is within 1 standard deviation of the mean P(7 < x < 23) =.68 = 15, = The Evaluation of Subjective Probabilities

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15-25 Elements of a decision analysis Elements of a decision analysis Actions Anything the decision-maker can do at any time Chance occurrences Possible outcomes (sample space) Probabilities associated with chance occurrences Final outcomes Payoff, reward, or loss associated with action Additional information Allows decision-maker to reevaluate probabilities and possible rewards and losses Decision Course of action to take in each possible situation Elements of a decision analysis Elements of a decision analysis Actions Anything the decision-maker can do at any time Chance occurrences Possible outcomes (sample space) Probabilities associated with chance occurrences Final outcomes Payoff, reward, or loss associated with action Additional information Allows decision-maker to reevaluate probabilities and possible rewards and losses Decision Course of action to take in each possible situation 15-5 Decision Analysis

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15-26 Market Do not market Productunsuccessful (P = 0.25) Productsuccessful (P = 0.75) $100,000 -$20,000 $0 DecisionDecision ChanceOccurrenceChanceOccurrenceFinalOutcomeFinalOutcome 15-6: Decision Tree: New-Product Introduction

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15-27 Product is ActionSuccessfulNot Successful Market the product$100,000 -$20,000 Do not market the product$0 $0 Product is ActionSuccessfulNot Successful Market the product$100,000 -$20,000 Do not market the product$0 $0 15-6: Payoff Table and Expected Values of Decisions: New-Product Introduction

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15-28 Market Do not market Product unsuccessful (P=0.25) Product successful (P=0.75) $100,000 -$20,000 $0 Expected Payoff $70,000 Expected Payoff $70,000 Expected Payoff $0 Expected Payoff $0 Nonoptimal decision branch is clipped Nonoptimal decision branch is clipped Clipping the Nonoptimal Decision Branches Solution to the New-Product Introduction Decision Tree

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15-29 OutcomePayoffProbability xP(x) Extremely successful$150, ,000 Very successful ,000 Successful100, ,000 Somewhat successful80, ,000 Barely successful40, ,000 Break even Unsuccessful-20, Disastrous-50, ,500 Expected Payoff: $77,500 OutcomePayoffProbability xP(x) Extremely successful$150, ,000 Very successful ,000 Successful100, ,000 Somewhat successful80, ,000 Barely successful40, ,000 Break even Unsuccessful-20, Disastrous-50, ,500 Expected Payoff: $77,500 New-Product Introduction: Extended-Possibilities

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15-30 Market Do not market $100,000 -$20,000 $0 Decision Chance Occurrence Chance Occurrence Payoff -$50,000 $0 $40,000 $80,000 $120,000 $150, Expected Payoff $77,500 Expected Payoff $77,500 Nonoptimal decision branch is clipped Nonoptimal decision branch is clipped New-Product Introduction: Extended-Possibilities Decision Tree

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15-31 $780,000 $750,000 $700,000 $680,000 $740,000 $800,000 $900,000 $1,000,000 Lease Not Lease Pr = 0.9 Pr = 0.1 Pr = 0.05 P r = 0.4 Pr = 0.6 Pr = 0.3 Pr = 0.15 Not Promote Promote Pr = 0.5 Example 15-3: Decision Tree

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15-32 $780,000 $750,000 $700,000 $680,000 $740,000 $800,000 $900,000 $1,000,000 Lease Not Lease Pr = 0.9 Pr = 0.1 Pr = 0.05 Pr = 0.4 Pr = 0.6 Pr = 0.3 Pr = 0.15 Not Promote Promote Expected payoff: $753,000 Expected payoff: $753,000 Expected payoff: $716,000 Expected payoff: $716,000 Expected payoff: $425,000 Expected payoff: $425,000 Expected payoff: $700,000 Expected payoff: $700,000 Pr=0.5 Expected payoff: 0.5* *716000= $783,000 Expected payoff: 0.5* *716000= $783,000 Example 15-3: Solution

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$100,000 $95,000 -$25,000 -$5,000 $95,000 -$25,000 -$5,000 -$20,000 Test Not test Test indicates success Test indicates failure Market Do not market Market Successful Failure Successful Failure Payoff Pr=0.25 Pr=0.75 New-Product Decision Tree with Testing New-Product Decision Tree with Testing 15-7 Handling Additional Information Using Bayes Theorem

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15-34 P(S)=0.75P(IS|S)=0.9P(IF|S)=0.1 P(F)=0.75P(IS|F)=0.15P(IF|S)=0.85 P(IS)=P(IS|S)P(S)+P(IS|F)P(F)=(0.9)(0.75)+(0.15)(0.25)= P(IF)=P(IF|S)P(S)+P(IF|F)P(F)=(0.1)(0.75)+(0.85)(0.25)= Applying Bayes Theorem

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$100,000 $95,000 -$25,000 -$5,000 $95,000 -$25,000 -$5,000 -$20,000 Test Not test P(IS)= Market Do not market Market P(S)=0.75 Payoff P(F)=0.25 P(IF)= P(S|IF)= P(F|IF)= P(S|IS)= P(F|IS)= $86,866 $6,308 $70,000 $6,308 $70,000 $ $70,000 Expected Payoffs and Solution

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15-36 Prior Information Level of Economic ProfitActivityProbability $3 millionLow 0.20 $6 millionMedium 0.50 $12 millionHigh 0.30 Prior Information Level of Economic ProfitActivityProbability $3 millionLow 0.20 $6 millionMedium 0.50 $12 millionHigh 0.30 Reliability of Consulting Firm Future State ofConsultants Conclusion EconomyHighMediumLow Low Medium High Reliability of Consulting Firm Future State ofConsultants Conclusion EconomyHighMediumLow Low Medium High Consultants say Low EventPriorConditionalJointPosterior Low Medium High P(Consultants say Low) Consultants say Low EventPriorConditionalJointPosterior Low Medium High P(Consultants say Low) Example 15-4: Payoffs and Probabilities

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15-37 Consultants say Medium EventPriorConditionalJointPosterior Low Medium High P(Consultants say Medium) Consultants say Medium EventPriorConditionalJointPosterior Low Medium High P(Consultants say Medium) Consultants say High EventPriorConditionalJointPosterior Low Medium High P(Consultants say High) Consultants say High EventPriorConditionalJointPosterior Low Medium High P(Consultants say High) Alternative Investment ProfitProbability $4 million 0.50 $7 million 0.50 Consulting fee: $1 million Alternative Investment ProfitProbability $4 million 0.50 $7 million 0.50 Consulting fee: $1 million Example 15-4: Joint and Conditional Probabilities

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15-38 Example 15-4: Decision Tree

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15-39 Dollars Utility Additional Utility Additional Utility Additional $1000 Additional Utility Additional $1000 } } { Utility is a measure of the total worth of a particular outcome. It reflects the decision makers attitude toward a collection of factors such as profit, loss, and risk. Utility is a measure of the total worth of a particular outcome. It reflects the decision makers attitude toward a collection of factors such as profit, loss, and risk Utility and Marginal Utility

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15-40 Utility Dollars Risk Averse Dollars Utility Risk Taker Utility Dollars Risk Neutral Dollars Mixed Utility Utility and Attitudes toward Risk

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15-41 PossibleInitialIndifference ReturnsUtilityProbabilitiesUtility $1, ,300(1500)(0.8)+(56000)(0.2)0.2 22,000 (1500)(0.3)+(56000)(0.7) ,000 (1500)(0.2)+(56000)(0.8) ,00011 PossibleInitialIndifference ReturnsUtilityProbabilitiesUtility $1, ,300(1500)(0.8)+(56000)(0.2)0.2 22,000 (1500)(0.3)+(56000)(0.7) ,000 (1500)(0.2)+(56000)(0.8) , Utility Dollars Example 15-5: Assessing Utility

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15-42 The expected value of perfect information (EVPI): EVPI = The expected monetary value of the decision situation when perfect information is available minus the expected value of the decision situation when no additional information is available. The expected value of perfect information (EVPI): EVPI = The expected monetary value of the decision situation when perfect information is available minus the expected value of the decision situation when no additional information is available. Expected Net Gain Sample Size Max n max Expected Net Gain from Sampling 15-9 The Value of Information

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15-43 $200 Fare $300 Fare Competitor:$200 Pr = 0.6 Competitor:$300 Pr = 0.4 Competitor:$300 Pr = 0.4 Competitor:$200 Pr = 0.6 $8 million $10 million $4 million $9 million Payoff Competitors Fare Competitors Fare Airline Fare Airline Fare Example 15-6: The Decision Tree

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15-44 If no additional information is available, the best strategy is to set the fare at $200 E(Payoff|200) = (.6)(8)+(.4)(9) = $8.4 million E(Payoff|300) = (.6)(4)+(.4)(10) = $6.4 million With further information, the expected payoff could be: E(Payoff|Information) = (.6)(8)+(.4)(10)=$8.8 million EVPI= = $.4 million If no additional information is available, the best strategy is to set the fare at $200 E(Payoff|200) = (.6)(8)+(.4)(9) = $8.4 million E(Payoff|300) = (.6)(4)+(.4)(10) = $6.4 million With further information, the expected payoff could be: E(Payoff|Information) = (.6)(8)+(.4)(10)=$8.8 million EVPI= = $.4 million Example 15-6: Value of Additional Information

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