$\frac{P(H_S|x)}{P(H_N|x)} = \Lambda(x) \frac{P(H_S)}{P(H_N)} = \Lambda(x) \frac{P(H_S)}{1-P(H_S)}$ 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.