x37
The Bayes strategy we found in the previous paragraph, s5,
would also take ai if t2 is observed (see table 2). In a similar wav
one can determine the Bayes actions when t3 or t2 is observed;
one finds in these cases that the Bayes action are a2 and a\. Summing
up, the Bayes strategy s5 («2, «1, «1) is found without having to
compute all possible strategies.
6. Testing hypotheses
In hypothesis testing, the set of possible states of nature 0 is
considered to consist of two disjoint sets C1 and C2. One formulates
a null hypothesis Ho and an alternative hypothesis Ha:
H0: 0£Ci
Ha -. eec2.
Ho is simple or composite according as C1 contains one or more
elements. Similarly for Ha and C2.
There are two available actions a\ and a2
ai accept H0
a2 reject H0
Rejecting H0 is usually considered to be equivalent with accepting
Ha «1 is optimal for 0 G Ci and a2 for 0 G C2. This means that the
regrets are
r(0,ai) 0 for 0GCi
r(Q,a2) 0 for 0 G C2
Let us further define
r(Q,a2) n for 0 G Ci
r(0,«i) m for 0 G C2
The variate to be observed is the test statistic which we will
denote by z. Let the set of its possible values be denoted by D.
Then a test is a strategy which takes a\ if the observed z is in some
subset D1 of D, and which takes a2 when z is not in £>1 but in D2,
the complement of Di with respect to D D2 is the critical region;
the choice of this critical region constitutes the strategy.
The action probabilities are
P{z G D( I 0 G Cj} i,j 1,2
We denote these action probabilities for the strategy s as follows:
P{z G Z)i I 0 G Ci} 1 as P{z G Di I 0 G C2} 1 (3S
P{z G £>2 I 0 G Ci} as P{z D2 I 0 G C2}
If 0 G C1, «i is the optimal action; it is taken if z G Di. If z G D2
we take a2 which is the wrong action in this case. There-
