  Even simply-defined, finite-state generators produce stochastic processes
that require tracking an uncountable infinity of probabilistic features for
optimal prediction. For processes generated by hidden Markov chains the
consequences are dramatic. Their predictive models are generically
infinite-state. And, until recently, one could determine neither their
intrinsic randomness nor structural complexity. The prequel, though, introduced
methods to accurately calculate the Shannon entropy rate (randomness) and to
constructively determine their minimal (though, infinite) set of predictive
features. Leveraging this, we address the complementary challenge of
determining how structured hidden Markov processes are by calculating their
statistical complexity dimension -- the information dimension of the minimal
set of predictive features. This tracks the divergence rate of the minimal
memory resources required to optimally predict a broad class of truly complex
processes.
