Relationship between Brier score and area under the binormal ROC curve

If we consider the Brier score (B) in the context of the signal detection theory and assume that it makes sense to consider the existence of B as a parameter for the population (let B be this B), and if we assume that the calibration in the observer's probability estimate is perfect, we find that there is a theoretical relationship between B and the area under the binormal receiver operating characteristic (ROC) curve, A(Z). We have derived this theoretical functional relationship between B and A(Z), by using the parameter a and b in the binormal ROC model and the prior probability of signal events (alpha); here, the two underlying normal distributions are N and N; and, a= and b=. We empirically found that, if parameters b and alpha are constant, B values in relation to given A(Z) values monotonically decrease as A(Z) values increase, and these relationship curves have monotonically decreasing slopes.