We put these results together in table 3, under which we repeat
for convenience table 2 in transposed form.
Now the expected losses can be computed. We remark that a
strategy is a function which assigns an action to an observation.
The observation is a realization of a random variable which has a
distribution dependent on 0. Consequently the action is a random
variable too: a s(t). We now compute the action probabilities,
i.e. for each strategy we find the probabilities to take ax or a2 when
0i is the state of nature and when 02 is the state of nature. We give
two examples of this computation:
The strategy si takes «1, whatever the observation may be.
When 0i is the state of nature, the action ai is taken with probability
1, «2 with probability zero. When 02 is the state of nature the action
«i is also taken with probability 1.
The strategy s2 takes ax if h or fa is observed. If 61 is the state
of nature, the probability of observing fa or fa is 0.3085 0.3830
0.691:5 (see table 3). If 02 is the state of nature, this probability is
°-9545 0.0428 0.9973. If k is observed, we take <z2. The proba
bility of observing fa is under 0X: 0.3085 and under 02: 0.0027.
The results for all strategies are given in table 4.
128
00
0
t2
8-12
t*
12
0,
02
0.3085
0.9545
0.3830
0.0428
0.3085
O.OO27
Sl
S 2
S3
*5
56
57
S«
dy
dy
«1
dy
a2
a2
a2
a2
ay
ay
«2
a2
ay
ay
a2
ax
a2
a2
ay
ay
a2
Table of strategies
TABLE 3.
Probability to observe the different values of \t\ if 0 is the state of nature
Si
ay
a2
0i
I
0
02
I
0
54
ay
a2
0i
0.6170
0.3830
02
0.9572
0.0428
S2
fll
«2
0i
O.69I5
O.3085
02
0-9973
6.0027
*5
ai
a2
01
02
0.6915
0-0455
0.3085
0-9545
S3
«1
a2
0i
O.3085
0.6915
02
0-9545
00455
S6
ai
«2
01
02
0.3830
O.O428
O.617O
O.9572
