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 entropy 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.