A Utility-Based Approach to Some Information Measures

We review a decision theoretic, i.e., utility-based, motivation for entropy and KullbackLeibler relative entropy, the natural generalizations that follow, and various properties of these generalized quantities. We then consider these generalized quantities in an easily interpreted special case. We show that the resulting quantities, share many of the properties of entropy and relative entropy, such as the data processing inequality and the second law of thermodynamics. We formulate an important statistical learning problem – probability estimation – in terms of a generalized relative entropy. The solution of this problem reflects general risk preferences via the utility function; moreover, the solution is optimal in a sense of robust absolute performance.