Upper bounds on class-wise sensitivities

The structural properties of the ordinal classifier cascade allow for the construction of upper limits on their empirical class-wise sensitivities. These bounds are based on the training of the cascade’s base classifiers and postulated in Theorem 1. Although this theorem is formulated for full cascades, the corresponding bounds can directly be applied for partial cascades.

Theorem 1 Let h denote an ordinal classifier cascade h:ℝn→Y={y(1),…,y(|Y|)} with base classifiers ℰ={c(1),…,c(|Y|−1)}. Let furthermore X(i) be a non-empty set of samples of class y_(i). Then the sensitivity of h for y_(i) is limited by

Proof. The theorem is a direct consequence of Lemmata 1 and 2 (see Supplementary).

Theorem 1 states that the sensitivities of an ordinal classifier cascade h can be upper bounded by several conditional prediction rates of its base classifiers. For class y_(i), the sensitivity of the cascade is limited by the corresponding sensitivity of its ith base classifier c_(i) (Eq. ⁴). It is also bounded by the predictions of all previous base classifiers c_(k), k < i (Eq. ⁵). A sample of class y_(i) will not be classified correctly, if it is classified as y_(k) by c_(k). The sensitivity of the cascade for class y_(i) is therefore also limited by the conditional prediction rate of c_(k) for predicting class label y_(k+1) for samples of class y_(i). A detailed theoretical proof can be found in the Supplementary.