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because the lowest Z.(02,s) equals zero (for s7). In the diagram of
fig. 2 this means that the convex set S of points representing
strategies is shifted until si is on the vertical axis. In the actual
diagram, the axis R(Qi,s) o is drawn as the vertical line through
Si. By constructing a square having its left bottom corner on the
new origin and enlarging it until it touches the boundary of the
convex set, we find s10, the minimax risk strategy. This strategy
chooses si and s5 with probabilities 0.141 and 0.859.
Serious criticism has been formulated e.g. in [4] against the
minimax risk criterion, because in certain cases a totally irrelevant
strategy, which will never be chosen, can influence the decision
materially. We will not go into this question.
Up to now, we have assumed that nothing was known about the
state of nature 0. The observation t was the only indication which
by its dependence on 0 provided information. In some cases we may
possess more information on 0, namely the probability with which
0 takes its different possible values0 is then considered as a random
variable. If the surveyor in our example knows from experience
that his team make a gross error of 10 cm in about 20% of their
distance measurements, he may attach the following a priori
probabilities to the possible states of nature:
P{0 0t} 0.2
P{0 02} 0.8
Adherents of the subjectivistic view of probability may express
their "degree of belief" by adopting a priori probabilities for the
different states of nature without having recourse to relative
frequencies. We will abstain from discussing this line of thought.
When a priori probabilities for the different states of nature are
known, a Bayes strategy can be computed. Let the probability of
01 be p and that of 02: 1 -p. Then a Bayes strategy, corresponding
to these a priori probabilities is a strategy which minimizes
L{s) p L(0i,s) (i-P) L(02,s)
To find Bayes strategies we must find a point of the set S, for
which this expression is a minimum. It is easily seen that this
point (or these points) are found in our example by drawing a
line in fig. 2 parallel to the line
p L(di,s) (1 -p) L(02,s) o
and moving this line parallel to itself until it touches the set S.
It has then become a "supporting line"; the point (or points) of
contact represent the Bayes strategy (strategies). The construction
has been carried out in fig. 2 for the a priori probabilities 0.2 and
0.8, giving the dot-dashed line supporting the convex set at s5
Consequently the Bayes strategy for these a priori probabilities
is s5. It will be noted that s5 is the Bayes strategy for a whole range
