0.1 Detection performance criteria  (Page 3/3)

This optimization problem can be solved using Lagrange multipliers ; we seek to find the decision rule that maximizes $$F=P_D-\lambda (P_F-\alpha ^\prime )$$ where $\lambda$ is a positive Lagrange multiplier. This optimization technique amounts to finding the decision rule that maximizes $F$ , then finding the value of the multiplier that allows the criterion toinge the detection probability in competition with false-alrm probabilitiesin excess of the criterion value. As is usual in the derivation of optimum decision rules, we maximize thesequantities with respect to the decision regions. Expressing $P_D$ and $P_F$ in terms of them, we have

We have not as yet found a value for the threshold. The false-alarm probability can be expressed in terms of theNeyman-Pearson threshold in two (useful) ways.

An important application of the likelihood ratio test occurs when $R$ is a Gaussian random vector for each model. Suppose the models correspond to Gaussian random vectorshaving different mean values but sharing the same covariance.

• ${ℳ}_0$ : $(R, N(0, \sigma ^{2}I))$
• ${ℳ}_1$ : $(R, N(m, \sigma ^{2}I))$

When trying to compute the probability of error or the threshold in the Neyman-Pearson criterion, we must find theconditional probability density of one of the decision statistics: the likelihood ratio, the log-likelihood, or thesufficient statistic. The log-likelihood and the sufficient statistic are quite similar in this problem, but clearly weshould use the latter. One practical property of the sufficient statistic is that it usually simplifiescomputations. For this Gaussian example, the sufficient statistic is a Gaussian random variable under each model.

• ${ℳ}_0$ : $(\Upsilon (r), N(0, L\sigma ^2))$
• ${ℳ}_1$ : $(\Upsilon (r), N(Lm, L\sigma ^2))$

To find the threshold for the Neyman-Pearson test from the expressions given on , we need the area under a Gaussian density.