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Decision Making UnderRisk Continued:

Bayes Theorem andPosterior Probabilities

Chapter 8

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Bayesian MethodsThere is a continuing debate among statisticians, littleknown to those outside the field, over the proper definition of probability . The frequentist definitionsees probability as the long-run expected frequency of occurrence . P(A) = n/N, where n is the number of times event A occurs in N opportunities . The Bay esi an

view of probability is related to degree of belief .

It is ameasure of the plausibility of an event givenincomplete knowledge .

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The Market Research Question

U rn 1

8 Red

2 White

P(Red/ U rn1) = 8/10 = 0.8(Known Population)

Have P(A/B)

U rn 2

? R

Wh a t is in the urn?

Select a samp le, andthen mak e inferencesabo ut the pop ul a tion(Unknown Population)

Want P(B/A)

Vs.

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How We Will Use Bayes' Theorem

Prior information can be based on the results of previous experiments, or expert opinion, and can be expressed as probabilities . If it is desirable toimprove on this state of knowledge, anexperiment can be conducted . Bayes' Theoremis the mechanism used to update the state of

knowledge with the results of the experiment to provide a posterior distribution .

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Bayes Theorem

Used to revise probabilities basedupon new data

Posterior probabilities

Prior probabilities

New data

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How Bayes' Theorem Works

Let the experiment be A and the prediction be B . Letsassume that both have occurred . The probability of both Aand B together is P(A B), or simply 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(A B) = P(B|A) * P(A)

Simple algebra shows that:P( B|A) = P(A |B ) * P( B) / P(A)

This is Bayes' Theorem .

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Ex ample Problem:Thompson Lumber Company

Thompson Lumber Company is trying to decidewhether to e x pand its product line bymanufacturing and marketing a new productwhich is backyard storage sheds.The courses of action that may be choseninclude:(1) large plant to manufacture storage sheds,(2) small plant to manufacture storage sheds, or(3) build no plant at all.

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Thompson Lumber Company

Probability 0.5 0.5

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Ex pected Monetary Value

Probability of favorable market is same asprobability of unfavorable market.Each state of nature has a 0.50 probability.

Th ompson Lumber Company

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Calculating the EVPI

Best outcome for state of nature "favorable market" is"build a large plant" with a payoff of $200,000.Best outcome for state of nature "unfavorablemarket" is "do nothing," with payoff of $0.

Therefore, Ex pected value with perfect informationEVwPI = ($200,000)(0.50) + ($0)(0.50) = $ 100,000If one had perfect information, an average payoff of $100,000 could be achieved in the long run.H owever, the ma x imum EMV (EVBES T) or e x pectedvalue without perfect information, is $40,000.Therefore, EVPI = $100,000 - $40,000 = $60,000.

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To Test or Not to TestOften, companies have the option to perform markettests/surveys, usually at a price, to get additionalinformation prior to making decisions.H owever, some interesting questions need to beanswered before this decision is made:

H ow will the test results be combined with priorinformation?

H ow much should you be willing to pay to test?The good news is that Bayes Theorem can be used to

combine the information, and we can use our decisiontree to find EVS I, the Ex pected Value of S ampleInformation.In order to perform these calculations, we first need toknow how reliable the potential test may be.

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Market Survey Reliability in PredictingActual S tates of Nature

Assuming that the above information is available, wecan combine these conditional probabilities with our

prior probabilities using Bayes Theorem .

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Ma rket Survey Reli ab ility in Predicting

Actua

l St a

tes of Na

ture

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Probability Revisions GivenPositive Survey

Alternatively, the following table will produce the same results:

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Probability Revisions Given

Negative Survey

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Placing Posterior Probabilities on theDecision Tree

The bottom of the tree is the no test part of the analysis; therefore, the prior probabilities are assigned to these events.P(favorable market) = P(FM) = 0.5P(unfavorable market) = P(UM) = 0.5

The calculations here will be identical to the EMV calculations performed without a decision tree.The top of the tree is the test part of the analysis; therefore, the posterior probabilities are assigned to these events.

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Decision Trees for Test/No TestMulti-stage Decision Problems

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Dec ision Tr ee SolutionTh ompson Lumber Company

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Ex pected Value of Sample Information

= EV of the top of the tree - EV of the bottom of the tree

This calculation ignores the cost of the test . Once youcompute the EVSI, you can compare it to the cost of the testto determine the desirability to test . The ratio of the EVSIto the EVPI times 100 will also give you a measure of efficiency of the proposed test, expressed as a percentage .

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Ex pected Value of Sample Information

For Thompson Lumber Company,

EVSI = $49,200 - $40,000 = $9,200And since the EVPI was previously calculated to be $60,000,Thompson would be willing to pay up to $9,200 for this test

information, with an efficiency of (9200/60000)*100 = 15.3%

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Decision Tree Class Ex ercise:Jenny Lind, Part 2 (see te x t, #8-16 and 8-37)

Jenny Lind may hire a market research firm to conduct asurvey at a cost of $100,000. The result of the surveywould be either a favorable (F) or unfavorable (U) publicresponse to the movie. The firm s ability to assess themarket is:P(F/S ) = 0.3 P(U/ S ) = 0.7P(F/M) = 0.6 P(U/M) = 0.4

P(F/L) = 0.8 P(U/L) = 0.2Should Jenny conduct the survey?What is the most she should be willing to pay for the

survey? What is the efficiency of this survey?