Bayesian structural inference for hidden processes

Christopher C. Strelioff and James P. Crutchfield
Phys. Rev. E 89, 042119 – Published 10 April 2014
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Abstract

We introduce a Bayesian approach to discovering patterns in structurally complex processes. The proposed method of Bayesian structural inference (BSI) relies on a set of candidate unifilar hidden Markov model (uHMM) topologies for inference of process structure from a data series. We employ a recently developed exact enumeration of topological ε-machines. (A sequel then removes the topological restriction.) This subset of the uHMM topologies has the added benefit that inferred models are guaranteed to be ε-machines, irrespective of estimated transition probabilities. Properties of ε-machines and uHMMs allow for the derivation of analytic expressions for estimating transition probabilities, inferring start states, and comparing the posterior probability of candidate model topologies, despite process internal structure being only indirectly present in data. We demonstrate BSI's effectiveness in estimating a process's randomness, as reflected by the Shannon entropy rate, and its structure, as quantified by the statistical complexity. We also compare using the posterior distribution over candidate models and the single, maximum a posteriori model for point estimation and show that the former more accurately reflects uncertainty in estimated values. We apply BSI to in-class examples of finite- and infinite-order Markov processes, as well to an out-of-class, infinite-state hidden process.

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  • Received 10 September 2013

DOI:https://doi.org/10.1103/PhysRevE.89.042119

©2014 American Physical Society

Authors & Affiliations

Christopher C. Strelioff1,* and James P. Crutchfield1,2,†

  • 1Complexity Sciences Center and Physics Department, University of California at Davis, One Shields Avenue, Davis, California 95616, USA
  • 2Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA

  • *strelioff@ucdavis.edu
  • chaos@ucdavis.edu

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Issue

Vol. 89, Iss. 4 — April 2014

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