0.1 Detection performance criteria

The criterion used in the previous section---minimize the average cost of an incorrect decision---may seem to be acontrived way of quantifying decisions. Well, often it is. For example, the Bayesian decision rule depends explicitly on the a priori probabilities. A rational method of assigning values to these---either by experiment or through trueknowledge of the relative likelihood of each model---may be unreasonable. In this section, we develop alternative decisionrules that try to respond to such objections. One essential point will emerge from these considerations: the likelihood ratio persists as the core of optimal detectors asoptimization criteria and problem complexity change . Even criteria remote fromperformance error measures can result in the likelihood ratio test. Such an invariance does not occur often in signal processing andunderlines the likelihood ratio test's importance.

Maximizing the probability of a correct decision

As only one model can describe any given set of data (the models are mutually exclusive), the probability of beingcorrect $P_c$ for distinguishing two models is given by $$P_c=(\text{say}{ℳ}_0\text{when}{ℳ}_0\text{true})+(\text{say}{ℳ}_1\text{when}{ℳ}_1\text{true})$$ We wish to determine the optimum decision region placement.Expressing the probability of being correct in terms of the likelihood functions $p(R, {ℳ}_i, r)$ , the a priori probabilities and the decision regions, we have $$P_c=\int {\pi }_{0}p(R, {ℳ}_0, r)\,d r+\int {\pi }_{1}p(R, {ℳ}_1, r)\,d r$$ We want to maximize $P_c$ by selecting the decision regions $Z_0$ and $Z_1$ . Mimicking the ideas of the previous section, we associate each value of $r$ with the largest integral in the expression for $P_c$ . Decision region $Z_0$ , for example, is defined by the collection of values of $r$ for which the first term is largest. As all of the quantities involved are non-negative, the decision rulemaximizing the probability of a correct decision is

To evaluate the quality of the decision rule, we usually compute the probability of error $P_e$ rather than the probability of being correct. This quantity can be expressed in terms of the observations, thelikelihood ratio, and the sufficient statistic.