Loss functions, complexities, and the Legendre transformation

The paper introduces a way of re-constructing a loss function from predictive complexity. We show that a loss function and expectations of the corresponding predictive complexity w.r.t. the Bernoulli distribution are related through the Legendre transformation. It is shown that if two loss functions specify the same complexity then they are equivalent in a strong sense. The expectations are also related to the so-called generalized entropy. c © 2003 Elsevier B.V. All rights reserved.