How to select an objective function using information theory

Science tests competing theories or models by evaluating the similarity of their predictions against observational experience. Thus, how we measure similarity fundamentally determines what we learn. In machine learning and scientific modeling, similarity metrics are used as objective functions. A classic example being mean squared error, which is the optimal measure of similarity when errors are normally distributed and independent and identically distributed (iid). In many cases, however, the error distribution is neither normal nor iid, so it is left to the scientist to determine an appropriate objective. Here, we review how information theory can guide that selection, then demonstrate the approach with a simple hydrologic model.