Confidence as posterior correctness probability ($P(\text{chosen answer correct}\mid E)$)

A normative Bayesian inference model of retrospective confidence. For two exhaustive alternatives with equal prior probabilities, let $L=\log [p(E\mid H_1)/p(E\mid H_0)]$ be the log-likelihood ratio. Bayes' theorem gives posterior odds $e^L$. Choosing the alternative with the larger posterior yields $P(\text{correct}\mid E)=(1+e^{-|L|}{)}^{-1}$. Unequal priors add prior log odds; choosing a different action changes which posterior probability is appropriate. This calculation does not establish that an observer knows the true likelihoods.