\[\frac{P(H_S|x)}{P(H_N|x)} = \Lambda(x) \frac{P(H_S)}{P(H_N)} = \Lambda(x) \frac{P(H_S)}{1-P(H_S)}\] Here we use the \(\Lambda\) most commonly used in SDT, which is the inverse of the likelihood ratio most often used in statistics.

It is natural to define the optimal criterion

\[ \Lambda(x) \frac{P(H_S)}{1-P(H_S)} = 1\] \[\beta^{*} = \Lambda(x) = \frac{1-P(H_S)}{P(H_S)}\] \[\lambda^{*} = log \left( \frac{1-P(H_S)}{P(H_S)} \right) = -\text{logit}(P(H_S))\] Observers shift the criterion in this direction when the proportion of signal trials is changed suggesting that they try to maximixe correct responses. The shift, however, is not perfect.