4.2 Performance evaluation

We alluded earlier to the relationship between the false-alarm probability $P_F$ and the detection probability $P_D$ as one varies the decision region. Because the Neyman-Pearson criterion depends on specifying the false-alarmprobability to yield an acceptable detection probability, we need to examine carefully how the detection probability isaffected by a specification of the false-alarm probability. The usual way these quantities are discussed is through a parametricplot of $P_D$ versus $P_F$ : the receiver operating characteristic or ROC.

As we discovered in the Gaussian example , the sufficient statistic provides the simplest way of computing theseprobabilities; thus, they are usually considered to depend on the threshold parameter $$ . In these terms, we have

We see that the detection probability is greater than or equal to the false-alarm probability. Since these probabilities mustdecrease monotonically as the threshold is increased, the ROC curve must be concave-down and must always exceed the equality line ( ).

The degree to which the ROC departs from the equality line $P_D=P_F$ measures the relative distinctiveness between the two hypothesized models for generating the observations. In the limit, the two models can be distinguishedperfectly if the ROC is discontinuous and consists of the point(1,0). The two are totally confused if the ROC lies on the equality line (this would mean, of course, that the two modelsare identical); distinguishing the two in this case would be "somewhat difficult".

Specification of the false-alarm probability for a new problem requires experience. Choosing a "reasonable" value for thefalse-alarm probability in the Neyman-Pearson criterion depends strongly on the problem difficulty. Too small a number willresult in small detection probabilities; too large and the detection probability will be close to unity, suggesting thatfewer false alarms could have been tolerated. Problem difficulty is assessed by the degree to which the conditionaldensities $p(r, {}_0, r)$ and $p(r, {}_1, r)$ overlap, a problem dependent measurement. If we are testing whether a distribution has one of two possible meanvalues as in our Gaussian example, a quantity like a signal-to-noise ratio will probably emerge as determiningperformance. The performance in this case can vary drastically depending on whether the signal-to-noise ratio is large orsmall. In other kinds of problems, the best possible performance provided by the likelihood ratio test can be poor.For example, consider the problem of determining which of two zero-mean probability densities describes a given set of dataconsisting of statistically independent observations (See this problem ). Presumably, the variances of these two densities are equal as weare trying to determine which density is most appropriate. In this case, the performance probabilities can be quite low,especially when the general shapes of the densities are similar. Thus a single quantity, like the signal-to-noise ratio, does not emerge to characterize problem difficulty in all hypothesis testing problems. In sequel, wewill analyze each model evaluation and detection problem in a standard way. After the sufficient statistic has been found, wewill seek a value for the threshold that attains a specified false-alarm probability. The detection probability will then bedetermined as a function of "problem difficulty", the measure of which is problem-dependent. We can control the choice offalse-alarm probability; we cannot control over problem difficulty. Confusingly, the detection probability will varywith both the specified false-alarm probability and the problem difficulty.

We are implicitly assuming that we have a rational method for choosing the false-alarm probability criterion value. In signalprocessing applications, we usually make a sequence of decisions and pass them to systems making more global determinations. Forexample, in digital communications problems the model evaluation formalism could be used to "receive" each bit. Each bit isreceived in sequence and then passed to the decoder which invokes error-correction algorithms. The important notions hereare that the decision-making process occurs at a given rate and that the decisions are presented to other signal processing systems. The rate at which errorsoccur in system input(s) greatly influences system design. Thus, the selection of a false-alarm probability is usuallygoverned by the error rate that can be tolerated by succeeding systems. If the decision rate is one per day, then amoderately large (say 0.1) false-alarm probability might be appropriate. If the decision rate is a million per second as ina one megabit communication channel, the false-alarm probability should be much lower: $10^{-12}$ would suffice for the one-tenth per day error rate.