A Bhattacharyya-type Conditional Error Bound for Quadratic Discriminant Analysis

Abstract

We give an upper bound on the conditional error of Quadratic Discriminant Analysis (QDA), conditioned on parameter estimates. In the case of maximum likelihood estimation (MLE), our bound recovers the well-known Chernoff and Bhattacharyya bounds in the infinite sample limit. We perform an empirical assessment of the behaviour of our bound in a finite sample MLE setting, demonstrating good agreement with the out-of-sample error, in contrast with the simpler but uninformative estimated error, which exhibits unnatural behaviour with respect to the sample size. Furthermore, our conditional error bound is applicable whenever the QDA decision function employs parameter estimates that differ from the true parameters, including regularised QDA.