Random Signals for Engineers using MATLAB and Mathcad
Copyright  1999 Springer Verlag NY
Example 7.7 Minimax Criteria
In the example we will derive a criteria for minimizing the cost or risk function when we have not selected
the apriori probability that the cost function is based upon. In this case we will first select a threshold based
on a given apriori probability and examine what happens to the cost function when the threshold remains
the same but the apriori probability changes.
Let us select a threshold as a function of the apriori probability or P(H1) = x for the case when d = 1. Select
a group of values for P(H1)
p1= 0.01:0.01:.99;
lam=log((1-p1)./p1)+1/2;
Risk function as a function of P(H1) from Equation 7.4-21 with c11 = c00 = 1 and c01 = c10 = 0 . Find the
threshold for P(H1) = .3 and fix the design
lam0=log((1-0.3)./0.3)+1/2
Risk0=(1-p1).*(1-cnorm(lam0,0,1))+p1*cnorm(lam0-1,0,1);
lam0 =
1.3473
lam1=log((1-0.5)./0.5)+1/2
Risk1=(1-p1).*(1-cnorm(lam1,0,1))+p1*cnorm(lam1-1,0,1);
lam1 =
0.5000
These are the Risk functions for an incorrect design point. We can now plot the Risk function for three
conditions.
Risk=(1-p1).*(1-cnorm(lam,0,1))+p1.*cnorm(lam-1,0,1);
plot(p1,Risk,p1,Risk1,p1,Risk0)
xlabel 'Probability'
ylabel 'Risk'
0.7
0.6
0.5
Risk
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Probability
0.7
0.8
0.9
1
We observe that when the risk function is chosen with an incorrect P(H1) the values can get as large as
0.61 for an incorrect value. A better choice would be with an L = 0.5 which given the minimum max
values for all values of p(H1) and is equal to 0.31. For this case it is the threshold values for a p(H1) of the
Risk curve where the slope is zero as shown in the figure.