5°
134
of a priori probabilities; the angle of slope of the line can vary a
great deal before another point of support is found. If the slope of
the line coincides with that of the line s5-s8, all strategies that are
mixtures of s6 and ss will be Bayes strategies. It can be shown that
every admissible strategy is a Bayes strategy for some a priori
probabilities p and l-pby letting p vary from o to i we get succes
sively for the Bayes strategys7, all points of the segment s7-s8, ss,
all points of sg-s6, s6, all points of s6-Sj and si, thus covering all
points of the admissible part of the boundary. By considering the
case of a convex set like the one in fig. 4 we see that a horizontal
supporting line there would give Si and S2 and their mixtures as
Bayes strategies for a priori probabilities 0 and 1clearly only si
is admissible, so that a Bayes strategy need not be admissible.
However, Bayes strategies for positive a priori probabilities p and
1 -p are admissible.
The Bayes strategy we found, s8, is the minimum expected loss
strategy. It is evident that the minimum risk strategy for the stated
a priori probabilities is the same strategy s6.
In this paragraph we have discussed minimax expected loss,
minimax risk and Bayes strategies. Bayes strategies can only be
found when a priori probabilities for the possible states of nature
are known. In cases where no a priori probabilities are known or
where the states of nature are not random, it may still be useful
to consider Bayes strategies in order to find admissible strategies.
Minimax strategies tend to protect against the largest losses or risks.
We have based our example on the losses given in table 1, which
were found by the fancy additions on page 8. Of course, in a con
crete case it would be very difficult to find an accurate evaluation
of the losses. To see how much the choice of a strategy is influenced
by variation of the losses (and, consequently, regrets), the example
has also been computed for the following table of regrets.
01
02
01
O
02
25
O
With these regrets, it is much more serious to take 02 (not
remeasure) if a gross error has been made. The risks become in
this case (losses have not been considered, to save space)
«1
£2
Ss
Si
s6
s.
S7 Sg
01
02
0
25
15-43
24-93
34-58
23.86
19-15
23-93
x5-43
1.14
20.85
I.O7
50 34-58
1.07 1 0.07
Fig. 3 has been constructed with these data; the Bayes strategy
is still s6. The minimax risk strategy is sn, not much different from
«10 in 2.
