Bayesian Statistics and Decision Analysis Session 10.

Bayesian Statistics and Decision Analysis

Session 10

• 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

• Summary and Review of Terms

15-1 Bayesian Statistics and Decision Analysis

ClassicalInference

DataStatistical Conclusion

Bayesian Inference

Data

PriorInformation

Statistical Conclusion

Bayesian statistical analysis incorporates a prior probability distribution and likelihoods of observed data to determine a posterior probability distribution of events.

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

• 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. .

P I( ) .0 001

P Z I P Z I( ) . ( ) . 92 08

( )Z I

( )Z I

P Z I P Z I( ) . ( ) . 0 04 0 96

Bayes’ Theorem: Example 10.1 (1)

P I

P I

P Z I

P Z I

( ) .

( ) .

( ) .

( ) .

0001

0999

092

004

P I ZP I Z

P Z

P I Z

P I Z P I Z

P Z I P I

P Z I P I P Z I P I

( )( )

( )

( )

( ) ( )

( ) ( )

( ) ( ) ( ) ( )

(. )( . )

(. )( . ) ( . )(. )

.

. .

.

..

92 0001

92 0001 004 999

000092

000092 003996

000092

040880225

Example 10.1: Applying Bayes’ Theorem

P I( ) .0001

P I( ) .0999 P Z I( ) .004

P Z I( ) .096

P Z I( ) .008

P Z I( ) .092 P Z I( ) ( . )( . ) . 0 001 0 92 00092

P Z I( ) ( . )( . ) . 0 001 0 08 00008

P Z I( ) ( . )( . ) . 0 999 0 04 03996

P Z I( ) ( . )( . ) . 0 999 0 96 95904

Prior Probabilities

Conditional Probabilities

JointProbabilities

Example 10.1: Decision Tree

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.

P xP x P

P x Pii

i

( )( ) ( )

( ) ( )

The likelihood function is the set of conditional probabilities P(x|) for given data x, considering a function of an unknown population parameter, .

The likelihood function is the set of conditional probabilities P(x|) for given data x, considering a function of an unknown population parameter, .

10-2 Bayes’ Theorem and Discrete Probability Models

Prior DistributionS P(S)0.1 0.050.2 0.150.3 0.200.4 0.300.5 0.200.6 0.10

1.00

LikelihoodBinomial with n = 20 and p = 0.100000 x P( X = x) 4.00 0.0898Binomial with n = 20 and p = 0.200000 x P( X = x) 4.00 0.2182Binomial with n = 20 and p = 0.300000 x P( X = x) 4.00 0.1304Binomial with n = 20 and p = 0.400000 x P( X = x) 4.00 0.0350Binomial with n = 20 and p = 0.500000 x P( X = x) 4.00 0.0046Binomial with n = 20 and p = 0.600000 x P( X = x) 4.00 0.0003

Example 10.2: Prior Distribution and Likelihoods of 4 Successes in 20 Trials

Prior PosteriorDistribution Likelihood DistributionS P(S) P(x|S) P(S)P(x|S) P(S|x) 0.1 0.05 0.0898 0.00449 0.060070.2 0.15 0.2182 0.03273 0.437860.3 0.20 0.1304 0.02608 0.348900.4 0.30 0.0350 0.01050 0.140470.5 0.20 0.0046 0.00092 0.012300.6 0.10 0.0003 0.00003 0.00040

1.00 0.07475 1.00000

Prior PosteriorDistribution Likelihood DistributionS P(S) P(x|S) P(S)P(x|S) P(S|x) 0.1 0.05 0.0898 0.00449 0.060070.2 0.15 0.2182 0.03273 0.437860.3 0.20 0.1304 0.02608 0.348900.4 0.30 0.0350 0.01050 0.140470.5 0.20 0.0046 0.00092 0.012300.6 0.10 0.0003 0.00003 0.00040

1.00 0.07475 1.00000

93% CredibleSet

Example 10.2: Prior Probabilities, Likelihoods, and Posterior Probabilities

0.60.50.40.30.20.1

0.5

0.4

0.3

0.2

0.1

0.0

S

P(S

)

P os te rio r D is tributio n o f Marke t S hare

0.60.50.40.30.20.1

0.5

0.4

0.3

0.2

0.1

0.0

S

P(S

)

P rio r D is tributio n o f Marke t S hare

Example 10.2: Prior and Posterior Distributions

Prior Distribution S P(S)0.1 0.060070.2 0.437860.3 0.348900.4 0.140470.5 0.012300.6 0.00040

1.00000

LikelihoodBinomial with n = 16 and p = 0.100000 x P( X = x) 3.00 0.1423Binomial with n = 16 and p = 0.200000 x P( X = x) 3.00 0.2463Binomial with n = 16 and p = 0.300000 x P( X = x) 3.00 0.1465Binomial with n = 16 and p = 0.400000 x P( X = x) 3.00 0.0468Binomial with n = 16 and p = 0.500000 x P( X = x) 3.00 0.0085Binomial with n = 16 and p = 0.600000 x P( X = x) 3.00 0.0008

Example 10.2: A Second Sampling with 3 Successes in 16 Trials

Prior PosteriorDistribution Likelihood Distribution S P(S) P(x|S) P(S)P(x|S) P(S|x) 0.1 0.06007 0.1423 0.0085480 0.0490740.2 0.43786 0.2463 0.1078449 0.6191380.3 0.34890 0.1465 0.0511138 0.2934440.4 0.14047 0.0468 0.0065740 0.0377410.5 0.01230 0.0085 0.0001046 0.0006010.6 0.00040 0.0008 0.0000003 0.000002

1.00000 0.1741856 1.000000

Prior PosteriorDistribution Likelihood Distribution S P(S) P(x|S) P(S)P(x|S) P(S|x) 0.1 0.06007 0.1423 0.0085480 0.0490740.2 0.43786 0.2463 0.1078449 0.6191380.3 0.34890 0.1465 0.0511138 0.2934440.4 0.14047 0.0468 0.0065740 0.0377410.5 0.01230 0.0085 0.0001046 0.0006010.6 0.00040 0.0008 0.0000003 0.000002

1.00000 0.1741856 1.000000

91% Credible Set

Example 10.2: Incorporating a Second Sample

Application of Bayes’ Theorem using Excel. The spreadsheet uses the BINOMDISTfunction in Excel to calculate the likelihood probabilities. The posterior probabilitiesare calculated using a formula based on Bayes’ Theorem for discrete random variables.

PRIOR DISTRIBUTION:S 0.1 0.2 0.3 0.4 0.5 0.6P(S) 0.05 0.15 0.2 0.3 0.02 0.1

LIKELIHOOD OF 4 OCCURRENCES IN 20 TRIALS GIVEN THE VALUES OF S:S 0.1 0.2 0.3 0.4 0.5 0.6P(x|S) 0.089778828 0.218199402 0.130420974 0.03499079 0.004620552 0.000269686

POSTERIOR DISTRIBUDTION AFTER THE FIRST SAMPLE:S 0.1 0.2 0.3 0.4 0.5 0.6P(S|x) 0.060051633 0.437850349 0.348946078 0.14042871 0.012362456 0.000360778

LIKEHOOD OF 3 OCCURRENCES IN 16 TRIALS GIVEN THE VALUES OF S:S 0.1 0.2 0.3 0.4 0.5 0.6P(x|S) 0.142344486 0.246290605 0.146496184 0.04680953 0.008544922 0.000811749

POSTERIOR DISTRIBUTION AFTER THE SECOND SAMPLE:S 0.1 0.2 0.3 0.4 0.5 0.6P(S|x) 0.049074356 0.619102674 0.293476795 0.03773804 0.00060646 1.68E-06

Example 10.2: Using Excel

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 for continuous distributions:

Total area under f x

f x f

f x f d

f x ff

( )( ) ( )

( ) ( )

( ) ( )( )

x

10-3 Bayes’ Theorem and Continuous Probability Distributions

• 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 posterior mean and variance of the normal population ofthe population mean, :

=

1 n

1 n

1 n

2 2

2 2 2 2

MM M

2 1

The Normal Probability Model

M n M s

M

M M

M

15 8 10 1154 684

2 1

815

6841154

8 684

2 1

8 684

=

12

n2

12

n2

12

n2

=

12

102

12

102

12

102

. .

..

. .

MM

= 11.77 Credible Set:

2 2 07795% 196 1177 196 2 077 7 699 15841

.. . ( . ) . [ . , . ]

The Normal Probability Model: Example 10.3

Likelihood

11.5411.77

PosteriorDistribution

PriorDistribution

15

Density

Example 10.3

• Based on normal distribution95% 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

10-4 The Evaluation of Subjective Probabilities

• Elements of a decision analysisActions

Anything the decision-maker can do at any timeChance occurrences

Possible outcomes (sample space)Probabilities associated with chance occurrencesFinal outcomes

Payoff, reward, or loss associated with actionAdditional information

Allows decision-maker to reevaluate probabilities and possible rewards and losses

Decision

Course of action to take in each possible situation

10-5 Decision Analysis

Market

Do notmarket

Productunsuccessful(P=0.25)

Productsuccessful(P=0.75)

$100,000

-$20,000

$0

DecisionDecisionChance Occurrence

Chance Occurrence

Final Outcome

Final Outcome

Decision Tree: New-Product Introduction

Product isAction Successful Not SuccessfulMarket the product $100,000 -$20,000Do not market the product $0 $0

The expected value of , denoted ( ):

= 750000 -5000 = 70,000

all x

X E XE X xP x

E Outcome

( ) ( )

( ) (100, )( . ) ( , )( . )

000 0 75 20 000 0 25

Payoff Table and Expected Values of Decisions: New-Product Introduction

Market

Do notmarket

Productunsuccessful(P=0.25)

Productsuccessful(P=0.75)

$100,000

-$20,000

$0

ExpectedPayoff$70,000


ExpectedPayoff$0

ExpectedPayoff$0

Nonoptimaldecision branchis clipped


Clipping the Nonoptimal Decision Branches

Solution to the New-Product Introduction Decision Tree

Outcome Payoff Probability xP(x)

Extremely successful $150,000 0.1 15,000Very successful 120.000 0.2 24,000Successful 100,000 0.3 30,000Somewhat successful 80,000 0.1 8,000Barely successful 40,000 0.1 4,000Break even 0 0.1 0Unsuccessful -20,000 0.05 -1000Disastrous -50,000 0.05 -2,500

Expected Payoff: $77,500

New-Product Introduction: Extended-Possibilities

Market

Do notmarket

$100,000

-$20,000

$0

DecisionChance Occurrence

Payoff

-$50,000

$0

$40,000$80,000

$120,000$150,000

0.2

0.3

0.05

0.1

0.1

0.1

0.1

0.05





New-Product Introduction: Extended-Possibilities Decision Tree

$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

PromotePr=0.5


$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




Pr=0.5Expected payoff: 0.5*425000+0.5*716000=$783,000

Example 10.4: Solution

0

$100,000

$95,000

-$25,000

-$5,000

$95,000

-$25,000

-$5,000

-$20,000

Test

Not test

Test indicatessuccess

Test indicatesfailure

Market

Do not market

Do not market

Do not market

Market

Market

Successful

Failure

Successful

Successful

Failure

Failure

Payoff

Pr=0.25

Pr=0.75

New-Product DecisionTree with Testing

New-Product DecisionTree with Testing

10-6 Handling Additional Information Using Bayes’ Theorem

0

$100,000

$95,000

-$25,000

-$5,000

$95,000

-$25,000

-$5,000

-$20,000

Test

Not test

P(IS)=0.7125

Market

Do not market

Do not market

Do not market

Market

Market

P(S)=0.75

Payoff

P(F)=0.25

P(IF)=0.2875

P(S|IF)=0.2609

P(F|IF)=0.7391

P(S|IS)=0.9474

P(F|IS)=0.0526

$86,866 $86,866

$6,308

$70,000

$6,308

$70,000

$66.003

$70,000

Expected Payoffs and Solution

Prior InformationLevel ofEconomic

Profit Activity Probability$3 million Low 0.20$6 million Medium 0.50$12 million High 0.30

Reliability of Consulting FirmFutureState of Consultants’ Conclusion Economy High Medium LowLow 0.05 0.05 0.90Medium 0.15 0.80 0.05High 0.85 0.10 0.05

Consultants say “Low” Event Prior Conditional Joint PosteriorLow 0.20 0.90 0.180 0.818Medium 0.50 0.05 0.025 0.114High 0.30 0.05 0.015 0.068 P(Consultants say “Low”) 0.220 1.000

Example 10.5: Payoffs and Probabilities

Consultants say “Medium” Event Prior Conditional Joint PosteriorLow 0.20 0.05 0.010 0.023Medium 0.50 0.80 0.400 0.909High 0.30 0.10 0.030 0.068 P(Consultants say “Medium”) 0.440 1.000

Consultants say “High” Event Prior Conditional Joint PosteriorLow 0.20 0.05 0.010 0.029Medium 0.50 0.15 0.075 0.221High 0.30 0.85 0.255 0.750 P(Consultants say “High”) 0.340 1.000

Alternative InvestmentProfit Probability$4 million 0.50$7 million 0.50

Consulting fee: $1 million

Example 10.5: Joint and Conditional Probabilities

$3 million

$6 million

$3 million

$11 million

$5 million

$2 million

$6 million

$3 million

$11 million

$5 million

$2 million

$6 million

$7 million

$4 million

$12 million

$6 million

$3 million

$11million

$5 million

$2 million

Hire consultantsDo not hire consultants

L

H L

H

L L L

MM M M

HHH

Invest Invest Invest InvestAlternative AlternativeAlternativeAlternative

0.5 0.5 0.50.5 0.5 0.5 0.50.5

0.3 0.2 0.5 0.750 0.221 0.029 0.068 0.909 0.023 0.068 0.114 0.818

5.5 7.2 4.54.54.5 9.413 5.339 2.954

M0.34

0.44

0.22

7.2 9.413 5.339 4.5

6.54


Application of Bayes’ Theorem to the information in Example 15-4 using Excel. The conditional probabilities of the consultants’ conclusion given the true future state and the prior distribution on the true future state are used to calculate the joint probabilities for each combination of true state and the consultants’ conclusion. The joint probabilities and Bayes’ Theorem are used to calculate the prior probabilities on the consultants’ conclusions and the conditional probabilities of the true future state given the consultants’ conclusion.

SEE NEXT SLIDE FOR EXCEL OUTPUT.


Distribution of Consultants' Conclusion Joint Probability of True StateGiven True Future State: and Consultants' Conclusion:

PriorCONSULTANTS' CONCLUSION Prob. Of CONSULTANTS' CONCLUSION

TRUE STATE "High" "Medium" "Low" True State TRUE STATE "High" "Medium" "Low"

Low 0.05 0.05 0.9 0.2 Low 0.01 0.01 0.18Medium 0.15 0.8 0.05 0.5 Medium 0.075 0.4 0.025High 0.85 0.1 0.05 0.3 High 0.255 0.03 0.015

CONSULTANTS' CONCLUSION

"High" "Medium" "Low"Prior Probability of Consultants' Conclusion: 0.34 0.44 0.22

CONSULTANTS' CONCLUSIONTRUE STATE "High" "Medium" "Low"

Posterior Distribution on True Future State Low 0.029 0.023 0.818Given the Consultants' Conclusion: Medium 0.221 0.909 0.114

High 0.75 0.068 0.068


Dollars

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 maker’s attitude toward a collection of factors such as profit, loss, and risk.

10-7 Utility and Marginal Utility

Utility

Dollars

Risk Averse

Dollars

Utility Risk Taker

Utility

Dollars

Risk Neutral

Dollars

MixedUtility

Utility and Attitudes toward Risk

Possible Initial IndifferenceReturns Utility Probabilities Utility$1,500 0 04,300 (1500)(0.8)+(56000)(0.2) 0.2

22,000 (1500)(0.3)+(56000)(0.7) 0.731,000 (1500)(0.2)+(56000)(0.8) 0.856,000 1 1

6000050000400003000020000100000

1.0

0.5

0.0

Utility

Dollars

Assessing Utility

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

nmax

Expected Net Gain from SamplingExpected Net Gain from Sampling

10-8 The Value of Information

$200Fare

$300Fare

Competitor:$200Pr=0.6




$8 million

$10 million

$4 million

$9 million

PayoffCompetitor’sFare

AirlineFare

8.4

6.4

Example 10.6: The Decision Tree

• 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 millionE(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=8.8-8.4 = $.4 million.

Example 10.6: Value of Additional Information

Bayesian Statistics and Decision Analysis Session 10.

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