3.8 Decision Analysis with ‘Unknown’ Information - Pre-posterior
Analysis
Often the decision maker has the possibility to ‘buy’ additional information through an
experiment before actually making her choice of action. If the cost of this information is small
in comparison to the potential value of the information, the decision maker should perform the
experiment. If several different types of experiments are possible the decision maker must
choose the experiment yielding the overall largest utility.
If the example from the previous sections is considered again the decision problem could be
formulated as a decision to decide whether or not to perform the trial pump tests.
The situation prior to performing the experiment has already been considered in Section 3.6.
There it was found that the expected cost based entirely on the prior information EC' is 70
monetary units.
In this situation the experiment is planned but the result is still unknown. In this situation the
expected cost, disregarding the experiment cost, can be found as:
1,...,
11''' 'min{''()nn
ii ijm j
iiEC P I E CI P I E Ca I
i } (3.7)
where is the number of different possible experiment findings and m is the number of
different decision alternatives. In Equation
n
(3.7) the only new term in comparison to the
previous section is PI'i which may be calculated by:
PI PI''ii :: : 11 P PI i 2 P': 2 (3.8)With reference to Section 3.6 and 3.7 the prior probabilities of obtaining the different
indications by the tests arePI' 1 , PI' 2 and PI' 3 given by:
PI PI' 1111122 :: ::P' PI P' 0.1 0.6 0.8 0.4 0.38PI' 2211222 PI ::P' PI ::P' 0.7 0.6 0.1 0.4 0.46PI PI' 3311322 ::P' PI ::P' 0.2 0.6 0.1 0.4 0.16The posterior expected cost in Equation (3.7) are found to be:
>?
>?
>?| 111 min [ | ] (100 10) [ 21 | ] 10; 100
min 0.158 110 0.842 10; 100
min 25.8;100 25.8@@ @@ @@
ECI P I P I
MU
::
>?
>?
>?
| 212 min [ | ] (100 10) [ 22 | ] 10; 100
min 0.913 110 0.087 10;100
min 101.3;100 100