Some unlikely properties of the likelihood ratio and its logarithm

摘要

Abstract: It is well known that the optimum way to perform a signal- detection or discrimination task is to compute the likelihood ratio and compare it to a threshold. Varying the threshold generates the receiver operating characteristic (ROC) curve, and the area under this curve (AUC) is a common figure of merit for task performance. AUC can be converted to a signal-to-noise ratio, often known as d$-a$/, using a well-known formula involving an error function. The ROC curve can also be determined by psychophysical studies for humans performing the same task, and again figures of merit such as AUC and d$-z$/ can be derived. Since the likelihood ratio is optimal, however, the d$-a$/ values for the human must necessarily be less than those for the ideal observer, and the square of the ratio of d$-a$/ (human)/d$-a$/(ideal) is frequently taken as a measure of the perceptual efficiency of the human. The applicability of this efficiency measure is limited, however, since there are very few problems for which we can actually compute d$-a$/ or AUC for the ideal observer. In this paper we examine some basic mathematical properties of the likelihood ratio and its logarithm. We demonstrate that there are strong constraints on the form of the probability density functions for these test statistics. In fact, if one knows, say, the density on the logarithm of the likelihood ratio under the null hypothesis, the densities of both the likelihood and the log-likelihood under both hypotheses are specified in terms of a likelihood-generating function. From this single function one can obtain all moments of both the likelihood and the log-likelihood under both hypotheses. Moreover, a AUC is expressed to an excellent approximation by a single point on the function. We illustrate these mathematical properties by considering the problem of signal detection with uncertain signal location. !7