P{0 0! 112) and P{0 02 I h]
135
5- The "No data" Problem
In certain decision problems, there is no observation variate.
We can imagine that in the surveyor's case of the last paragraph,
he is able to establish his tables of losses and regrets, but is unable
to establish a connection between the state of nature and an obser
vation variate. This would, e.g., be the case if his standard practice
were to measure every distance only once without further checking
possibilities. His losses and regrets are again:
«1
«2
ai
Clo
01
02
20
37 0i
0
17
25
O 02
25
0
It is possible in this case of a "no data" problem to indicate a
minimum loss action («1) and a minimum regret action (a2), but
not a minimax expected loss and a minimax risk action. If, however,
he knows the a priori probabilities for 0i and 82, say 0.2 and 0.8,
then he can compute his risks:
R(ai) 0.2 X 0 0.8 x 25 20
R(a2) 0.2 X 17 0.8 x o 3.4
The action a2 has minimum risk, so that this is the Bayes action
in the "no data" problem. (A similar computation shows that a2
has also minimum expected loss.)
If we do have an experiment at our disposal to estimate the
state of nature, and besides know a priori probabilities for the states
of nature, like was supposed in the previous paragraph, it is not
necessary to compute all possible strategies and then construct a
supporting line to find the Bayes strategy. We can reduce the
situation to a "no data" problem by the following consideration:
The observation variate t has a distribution which is dependent
on 0. 0 has a certain a priori distribution. We can now determine
the probability that 0i is the state of nature, it being given that t
assumes the observed value, say t2, and the same for 02. In other
words we can compute the conditional probabilities:
These probabilities define the a posteriori distribution, which
can now be used to determine the minimum risk strategy from the
"no data" problem that has been created by "digesting" into the
probabilities of 0i and 02 the information contained in the obser
vation.
We illustrate this with the example treated before. Let the
observation be t2, i.e. 8 \t\ 12. The a priori probabilities for
0i and 02 are 0.2 and 0.8.
We are going to compute
P{0 0i 1t2} or, shorter, P{0i fa)
