Ideal observers are mathematical models derived for the detection of particular types of stimuli under particular conditions. The ideal observer has as input a well-defined stimulus set and has as an output some evidence, analogous to a test statistic. The ideal observer transforms the stimulus set into the evidence, which is then used to make a decision about which event occurred. ROC analysis follows to measure the performance of the Ideal Observer.

When an ideal observer is derived mathematically, it is derived for all possible signals of a particular type. For instance, Gaussian noise with a bandwidth of 50 Hz and a duration of 20 msecs. Because Gaussian noise is random, there is an infinite number of signals that fit this description.

The evidence output by the ideal observer will take the form of a probability distribution of all the possible values of the evidence conditional on each event. The values of the evidence form a decision axis. These probability distributions are used to define the theoretical ROC curve.

The ROC curve and the area define the performance of the ideal observer. This theoretical performance may be of interest in its own right but more often than not is used as a model of some real detection process. It gives a benchmark to compare real performance.