Appendix A. Proof of deterministic overconfidence (binary classification) - AWS Prescriptive Guidance

Appendix A. Proof of deterministic overconfidence (binary classification)

Consider the softmax (logistic) function binary classification:

Softmax function binary classification

The softmax function is concave for Mathematical notation showing u as an element of the set of real numbers. and convex for Mathematical notation showing u as an element of the set of real numbers. . Therefore, by Jensen's inequality:

Jensen's inequality

This implies Mathematical inequality comparing H of p to the power of D with H of p to the power of B. for all Mathematical expression showing u is an element of the set of real numbers. . Equality is obtained when Mathematical equation showing p(D) equals p(B) equals 0.5. .