Properties of the Statistical Complexity Functional and Partially Deterministic HMMs

Abstract: Statistical complexity is a measure of complexity of discrete-time stationary stochastic processes, which has many applications. We investigate its more abstract properties as a non-linear function of the space of processes and show its close relation to the Knight’s prediction process. We prove lower semi-continuity, concavity, and a formula for the ergodic decomposition of statistical complexity. On the way, we show that the discrete version of the prediction process has a continuous Markov transition. We also prove that, given the past output of a partially deterministic hidden Markov model (HMM), the uncertainty of the internal state is constant over time and knowledge of the internal state gives no additional information on the future output. Using this fact, we show that the causal state distribution is the unique stationary representation on prediction space that may have finite entropy.