Single point estimation of a decision space

Signal detection theory (SDT) was developed to provide independent measures of sensitivity and bias for an observer asked to discriminate a signal stimulus against background noise. The sensitivity measure, $d^{\text{'}}$ , achieves this goal when the underlying decision space consists of two Gaussian distributions of equal variance. However, $d^{\text{'}}$ fails to provide a stable measure of sensitivity when the distributions are of unequal variance. In addition, unequal base rates of stimulus presentations or asymmetries in the payoff matrix for decision outcomes further shift criterion placement and estimations of sensitivity. Alternative sensitivity metrics that attempt to consider these scenarios either require information across multiple confidence levels or make implicit assumptions about the underlying decision space a priori. I propose an optimization approach that accurately estimates information about the underlying decision space without requiring information over multiple confidence levels. The proposed approach requires $\beta$ and a single false-alarm and hit rate pair. The reliance on $\beta$ limits the proposed method to providing a normative model of performance where the researcher is operating under a theoretical framework of decision. Simulations illustrate that in cases where $\beta$ is known, or can be reasonably estimated, the optimization approach is successful in recovering the critical characteristics of the decision space.