A Bayesian interpretation of the "proper" binormal ROC model using a uniform prior distribution for the area under the curve

Maximum likelihood estimation of receiver operating characteristic (ROC) curves using the "proper" binormal model can be interpreted in terms of Bayesian estimation as assuming a flat joint prior distribution on the c and d _a parameters However, this is equivalent to assuming a non-flat prior distribution for the area under the curve (AUC) that peaks at AUC - 1.0. We hypothesize that this implicit prior on AUC biases the maximum likelihood estimate (MLE) of AUC We propose a Bayesian implementation of the "proper" binomial ROC curve-fitting model with a prior distribution that is marginally flat on AUC and conditionally flat over c. This specifies a non-flat joint prior for c and d _a We developed a Monte Carlo Markov chain (MCMC) algorithm to estimate the posterior distribution and the maximum a posteriori (MAP) estimate of AUC. We performed a simulation study using 500 draws of a small dataset (25 normal and 25 abnormal cases) with an underlying AUC value of 0.85. When the prior distribution was a flat joint prior on c and d _a the MLE and MAP estimates agreed, suggesting that the MCMC algorithm worked correctly. When the prior distribution was marginally flat on AUC, the MAP estimate of AUC appeared to be biased low. However the MAP estimate of AUC for perfectly separable degenerate datasets did not appear to be biased. Further work is needed to validate the algorithm and refine the prior assumptions.