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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
24/2009

Properties of the Statistical Complexity Functional and Partially Deterministic HMMs

Wolfgang Löhr

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 functional on the space of processes and show its close relation to 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.

Received:
15.06.09
Published:
16.06.09
Keywords:
statistical complexity, prediction process, lower semi-continuity, ergodic decomposition, concavity, partially deterministic HMM

Related publications

inJournal
2009 Journal Open Access
Wolfgang Löhr

Properties of the statistical complexity functional and partially deterministic HMMs

In: Entropy, 11 (2009) 3, pp. 385-401