The likelihood ratio is the theoretically optimal decision variable for detection decisions. For any observation x on the internal evidence axis, the likelihood ratio β(x) is the ratio of the probability density of x under the signal-present distribution to the probability density under the noise-alone distribution. The Neyman-Pearson lemma proves that a decision rule based on the likelihood ratio is optimal in the sense that no other rule achieves a higher hit rate for the same false alarm rate.
Optimal Decision Rule
Optimal criterion: β* = [P(N)/P(S)] × [C(FA) − C(CR)] / [C(Hit) − C(Miss)]
For equal-variance Gaussian SDT:
β(x) = exp(d′·x − d′²/2)
Connection to Bayesian Decision Theory
The likelihood ratio framework connects SDT directly to Bayesian decision theory. The posterior probability that the signal is present is P(S|x) = β(x)·P(S) / [β(x)·P(S) + P(N)]. An ideal Bayesian observer computes the posterior odds — the product of the likelihood ratio and the prior odds — and responds "signal" when the posterior odds exceed a threshold determined by the costs and benefits of each outcome.
In practice, human observers do not perfectly compute likelihood ratios, but their behavior is often well approximated by a monotone function of the likelihood ratio with criterion settings that respond systematically (if imperfectly) to changes in signal probability and payoffs.