A Mathematical Framework for Quantifying Predictability Through Relative Entropy

Abstract

Kleeman has recently demonstrated that the relative entropy provides a significant measure of the information content of a prediction ensemble compared with the climate record in several simplified climate models. Here several additional aspects of utilizing the relative en-tropy for predictability theory are developed with full mathematical rigor in a systematic fashion which the authors believe will be very useful in practical problems with many degrees of freedom in atmosphere/ocean and biological science. The results developed here include a generalized signal-dispersion decomposition, rigorous explicit lower bound estimators for information content, and rigorous lower bound estimates on relative entropy for many variables, N , through N , one-dimensional relative entropies and N , two-dimensional mutual information functions. These last results provide a practical context for rapid evaluation of the predictive information content in a large number of variables.