Transcript Chapter08slides

Chapter 8
Revising Judgments
in the Light of
New Information
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
In this chapter we will look at the process
of revising initial probability estimates in
the light of new information.
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Bayes’ theorem
Prior probability
New information
Posterior probability
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The components problem (Fig. 8.1)
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In total, we would expect 410 (i.e. 140 +
270) components to fail the test.
 Now the component you selected is one of
these 410 components. Of these, only 140
are 'OK7, so your posterior probability that
the component is 'OK7 should be 140/410,
which is 0.341, i.e.
P(component OK|failed test) = 140/410 =
0.341
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Applying Bayes’ theorem to the components
problem (Fig. 8.2)
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The steps in the process which we have just applied are summarized
below:
(1) Construct a tree with branches representing all the possible events
which can occur and write the prior probabilities for these events on
the branches.
(2) Extend the tree by attaching to each branch a new branch which
represents the new information which you have obtained. On each
branch write the conditiona1 probability of obtaining this information
given the circumstance represented by the preceding branch.
(3) Obtain the joint probabilities by multiplying each prior probability
by the conditional probability which follows it on the tree.
(4) Sum the joint probabilities.
(5) Divide the 'appropriate' joint probability by the sum of the joint
probabilities to obtain the required posterior probability.
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Example:
An engineer makes a cursory inspection of a piece of
equipment and estimates that there is a 75% chance that it
is running at peak efficiency. He then receives a report that
the operating temperature of the machine is exceeding 80°
C. Past records of operating performance suggest that there
is only a 0.3 probability of this temperature being exceeded
when the machine is working at peak efficiency. The
probability of the temperature being exceeded if the
machine is not working at peak efficiency is 0.8. What
should be the engineer's revised probability that the
machine is operating at peak efficiency?
 Refer to Fig. 8.3
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Another example
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(more than two events )
A company's sales manager estimates that there is
a 0.2 probability that sales in the coming year will
be high, a 0.7 probability that they will be medium
and a 0.1 probability that they will be low. She
then receives a sales forecast from her assistant
and the forecast suggests that sales will be high.
By examining the track record of the assistant's
forecasts she is able to obtain the following
probabilities:
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p(high sales forecast given that the market
will generate high sales) = 0.9
p(high sales forecast given that the market
will generate only medium sales) =0.6
p(high sales forecast given that the market
will generate only low sales) = 0.3
Refer to Fig. 8.4
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We obtain the following posterior
probabilities:
p(high sales) = 0.18/0.63 = 0.2857
P(medium sales) = 0.42/0.63 = 0.6667
p(low sales) = 0.03/0.63 = 0.0476
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The effect of new information on the
revision of probability judgments
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It is interesting to explore the relative
influence which prior probabilities and new
information have on the resulting posterior
probabilities.
Consider that a situation where the geologist
is not very confident about his prior
probabilities and where the test drilling is
very reliable.
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Vague priors and very reliable information
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The posterior probabilities depend only
upon the reliability of the new information.
The 'vague' prior probabilities have had
no influence on the result.
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A more general view of the relationship
between the 'vagueness' of the prior
probabilities and the reliability of the new
information can be seen in Figure 8.6.
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The effect of
the reliability
of information
on the
modification
of prior
probabilities
(+)
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If the test drilling has only a 50%
probability of giving a correct result then its
result will not be of any interest and the
posterior probability will equal the prior, as
shown by the diagonal line on the graph.
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The more reliable the new information, the
greater will be the modification of the prior
probabilities.
For any given level of reliability, however,
this modification is relatively small either
where the prior probability is high, or
where the prior probability is very small.
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At the extreme, if your prior probability of
an event occurring is zero then the
posterior probability will also be zero.
In general, assigning prior probabilities of
zero or one is unwise.
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Applying Bayes’ theorem to a
decision problem
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Decision
Hold small stocks
Hold large stocks
Low sales
High sales
$80000
$20000
$140000
$220000
Profit $20000 $80000 $140000
Utility
0
0.5
0.8
$220000
1.0
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The retailer estimates that there is a 0.4
probability that sales will be low and a 0.6
probability that they will be high.
What level of stocks should he hold?
In Figure 8.7(a), It can be seen that his
expected utility is maximized if he decides to
hold a small stock of the commodity.
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The retailer’s problem with prior
probabilities
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Before implementing his decision the retailer
receives a sales forecast which suggests that
sales will be high.
P(forecast of high sales|high sales)=0.75
P(forecast of high sales|high sales)=0.2
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Applying Bayes’ theorem to the retailer’s
problem
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Applying posterior probabilities to the retailer’s
problem
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Assessing the value of new
information
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New information can remove or reduce the
uncertainty involved in a decision and
thereby increase the expected payoff.
Whether it is worth obtaining the
information in the first place or, if there are
several potential sources of information,
which one is to be preferred.
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The expected value of perfect
information
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The concept of the expected value of perfect
information (EVPI) can still be useful.
A problem is used to show how the value of
perfect information can be measured.
For simplicity, we will assume that the
decision maker is neutral to risk so that the
expected monetary value criterion can be
applied.
Refer to the following figure. (Descriptions
are in page 227)
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Determining the EVPI (Fig. 8.8)
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Calculating the EVPI
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If the test is perfectly accurate it would not
be worth paying them more than $15 000.
It is likely that the test will be less than
perfect, in which case the information it
yields will be of less value. Nevertheless, the
EVPI can be very useful in giving an upper
bound to the value of new information.
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If the manager is risk averse or risk seeking
or if he also has non-monetary objectives
then it may be worth him paying more or
less than this amount.
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The expected value of imperfect
information
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Suppose that, after making further enquiries, the
farm manager discovers that the Ceres test is not
perfectly reliable.
If the virus is still present in the soil the test has
only a 90% chance of detecting it, while if the
virus has been eliminated there is a 20% chance
that the test will incorrectly indicate its presence.
How much would it now be worth paying for the
test?
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Deciding whether to buy imperfect information
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If test indicates virus is present
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If test indicates virus is absent
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Determining the EVII
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Expected profit with imperfect information = $62 155
Expected profit without the information
Expected value of imperfect information
(EVII)
= $57 000
= $5 155
Refer to Page 232
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It would not, therefore, be worth paying
Ceres more than $5155 for the test.
You will recall that the expected value of
perfect information was $15 000, so the
value of information from this test is much
less than that from a perfectly reliable test.
Of course, the more reliable the new
information, the closer its expected value
will be to the EVP1.
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A summary of the main stages
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(1) Determine the course of action which would be chosen
using only the prior probabilities and record the expected
payoff of this course of action;
(2) Identify the possible indications which the new
information can give;
(3) For each indication:
(a) Determine the probability that this indication will
occur;
(b) Use Bayes' theorem to revise the probabilities in the
light of this indication;
(c) Determine the best course of action in the light of this
indication (i.e. using the posterior probabilities) and the
expected payoff of this course of action;
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(4) Multiply the probability of each indication
occurring by the expected payoff of the course of
action which should be taken if that indication
occurs and sum the resulting products. This will
give the expected payoff with imperfect
information;
(5) The expected value of the imperfect
information is equal to the expected payoff with
imperfect information (derived in stage 4) less the
expected payoff of the course of action which
would be selected using the prior probabilities
(which was derived in stage 1).
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