  Scientific explanation often requires inferring maximally predictive features
from a given data set. Unfortunately, the collection of minimal maximally
predictive features for most stochastic processes is uncountably infinite. In
such cases, one compromises and instead seeks nearly maximally predictive
features. Here, we derive upper-bounds on the rates at which the number and the
coding cost of nearly maximally predictive features scales with desired
predictive power. The rates are determined by the fractal dimensions of a
process' mixed-state distribution. These results, in turn, show how widely-used
finite-order Markov models can fail as predictors and that mixed-state
predictive features offer a substantial improvement.
