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When memory pays: Discord in hidden Markov models.

Emma Lathouwers1, John Bechhoefer2

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Memory is beneficial when analyzing systems with noisy data. Hidden Markov models reveal a critical point where using past observations significantly improves state estimation accuracy.

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Area of Science:

  • Complex systems
  • Statistical modeling
  • Information theory

Background:

  • Understanding when to retain historical data is crucial for accurate system analysis.
  • Noisy observations complicate state estimation in discrete systems.
  • Hidden Markov models (HMMs) are widely used for state inference.

Purpose of the Study:

  • To determine the conditions under which incorporating memory of past observations improves state estimation in HMMs.
  • To identify phase transitions related to the value of memory in noisy systems.
  • To analytically derive the critical point where memory becomes advantageous.

Main Methods:

  • Utilized hidden Markov models (HMMs) with varying parameters (states, symbols, transition symmetry).
  • Compared state estimation using naive observations (current data only) versus Bayesian filtering (historical data).
  • Defined and employed a discord order parameter to differentiate state estimates.
  • Mapped HMMs to Ising models to analyze phase transitions.

Main Results:

  • Identified phase transitions in state estimation accuracy based on memory usage.
  • Found that all explored HMM configurations exhibited similar behavior regarding phase transitions.
  • Analytically calculated the critical point where memory retention becomes beneficial.
  • The mapping to Ising models provided insights into the nature of these phase transitions.

Conclusions:

  • A quantifiable critical point exists where memory in observations transitions from being unhelpful to beneficial.
  • The study provides a framework for understanding the trade-offs between computational cost and accuracy in state estimation.
  • Phase transitions in HMMs offer a valuable lens for analyzing information processing in complex systems.