Risk analysis, ideal observers, and receiver operating characteristic curves for tasks that combine detection and estimation

Previously published work on joint estimation/detection tasks has focused on the area under the estimation receiver operating characteristic (EROC) curve as a figure of merit (FOM) for these tasks in imaging. Another FOM for these joint tasks is the Bayesian risk, where a cost is assigned to all detection outcomes and to the estimation errors, and then averaged over all sources of randomness in the object ensemble and the imaging system. Important elements of the cost function, which are not included in standard EROC analysis, are that the cost for a false positive depends on the estimate produced for the parameter vector, and the cost for a false negative depends on the true value of the parameter vector. The ideal observer in this setting, which minimizes the risk, is derived for two applications. In the first application, a parameter vector is estimated only in the case of a signal present classification. For the second application, parameter vectors are estimated for either classification, and these vectors may have different dimensions. In both applications, a risk-based estimation receiver operating characteristic curve is defined and an expression for the area under this curve is given. It is also shown that, for some observers, this area may be estimated from a two alternative forced choice test. Finally, if the classifier is optimized for a given estimator, then it is shown that the slope of the risk-based estimation receiver operating characteristic curve at each point is the negative of the ratio of the prior probabilities for the two classes.