Some Useful Properties of the Bayes Risk in Classification

In this article we study some properties of the Bayes risk with respect to prior probabilities in the context of two class classification problems. We show that Bayes risk is maximized when the ratio of the prior probabilities equals the median of the likelihood ratio statistic under the average density. We also show that when probability density functions (pdf) are symmetric and differ only in locations and the likelihood ratio is monotonic, the Bayes risk has a unique maximum at prior probability , and decreases as the difference between the prior probabilities increases. Several interesting examples are cited to illustrate the results.