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Expected Value of Sample Information

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Decision Making Under
Risk Continued:
Bayes’Theorem and
Posterior Probabilities
MGS3100 - Chapter 8
Slides 8c
Bayesian Methods
There is a continuing debate among statisticians, little
known to those outside the field, over the proper
definition of probability. The frequentist definition
sees 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 Bayesian
view of probability is related to degree of belief. It is a
measure of the plausibility of an event given
incomplete knowledge.
The Market Research Question
P(Red/Urn1) = 8/10 = 0.8
8 Red
(Known Population)
2 White
Urn 1
Have P(A/B)
Vs.
What is in the urn?
?
Select a sample, and
then make inferences
about the population
Urn 2
R
(Unknown Population)
Want P(B/A)
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 to
improve on this state of knowledge, an
experiment can be conducted. Bayes' Theorem
is the mechanism used to update the state of
knowledge with the results of the experiment to
provide a posterior distribution.
Bayes’ Theorem
Used to revise probabilities based
upon new data
Prior
probabilities
New data
Posterior
probabilities
How Bayes' Theorem Works
Let the experiment be A and the prediction be B. Let’s
assume that both have occurred. The probability of both A
and 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(AB) = P(B|A) * P(A)
Simple algebra shows that:
P(B|A) = P(A|B) * P(B) / P(A)
This is Bayes' Theorem.
Example Problem:
Thompson Lumber Company
Thompson Lumber Company is trying to decide
whether to expand its product line by
manufacturing and marketing a new product
which is “backyard storage sheds.”
The courses of action that may be chosen
include:
(1) large plant to manufacture storage sheds,
(2) small plant to manufacture storage sheds, or
(3) build no plant at all.
Thompson Lumber Company
Probability
0.5
0.5
Expected Monetary Value
Thompson Lumber Company
• Probability of favorable market is same as
probability of unfavorable market.
• Each state of nature has a 0.50 probability.
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 "unfavorable market"
is "do nothing," with payoff of $0.
• Therefore, Expected value with perfect information
EVwPI = ($200,000)(0.50) + ($0)(0.50) = $ 100,000
• If one had perfect information, an average payoff of
$100,000 could be achieved in the long run.
• However, the maximum EMV (EVBEST) or expected
value without perfect information, is $40,000.
• Therefore, EVPI = $100,000 - $40,000 = $60,000.
To Test or Not to Test
• Often, companies have the option to perform market
tests/surveys, usually at a price, to get additional
information prior to making decisions.
• However, some interesting questions need to be
answered before this decision is made:
– How will the test results be combined with prior information?
– How 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 decision
tree to find EVSI, the Expected Value of Sample
Information.
• In order to perform these calculations, we first need to
know how reliable the potential test may be.
Market Survey Reliability in Predicting
Actual States of Nature
Assuming that the above information is available, we
can combine these conditional probabilities with our
prior probabilities using Bayes’ Theorem.
Market Survey Reliability in
Predicting Actual States of Nature
Probability Revisions Given
Positive Survey
Alternatively, the following table will produce the same results:
Probability Revisions Given
Negative Survey
Placing Posterior Probabilities on the
Decision 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) = 0.5
P(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.
Decision Trees for Test/No Test
Multi-stage Decision Problems
Decision Tree Solution
Thompson Lumber Company
Expected 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 you
compute the EVSI, you can compare it to the cost of the test
to determine the desirability to test. The ratio of the EVSI
to the EVPI times 100 will also give you a measure of
“efficiency” of the proposed test, expressed as a percentage.
Expected Value of Sample Information
For Thompson Lumber Company,
EVSI = $49,200 - $40,000 = $9,200
And 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%
Decision Tree Class Exercise:
Jenny Lind, Part 2 (see text, #8-16 and 8-37)
Jenny Lind may hire a market research firm to conduct a
survey at a cost of $100,000. The result of the survey
would be either a favorable (F) or unfavorable (U)
public response to the movie. The firm’s ability to
assess the market is:
P(F/S) = 0.3
P(U/S) = 0.7
P(F/M) = 0.6
P(U/M) = 0.4
P(F/L) = 0.8
P(U/L) = 0.2
•Should Jenny conduct the survey?
•What is the most she should be willing to pay for the
survey? What is the efficiency of this survey?
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