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David B Saakian1

  • 1Theoretical Physics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam; Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam; and A.I. Alikhanyan National Science Laboratory Foundation, Yerevan Physics Institute, 2 Alikhanian Brothers Street, Yerevan 375036, Armenia.

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We present a new method for solving hidden Markov processes (HMPs) by mapping them to functional equations. This approach yields an exact entropy expression for HMPs, offering an alternative to existing integral equation solutions.

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Area of Science:

  • Statistical Physics
  • Information Theory
  • Stochastic Processes

Background:

  • Hidden Markov Processes (HMPs) are widely used to model systems with unobserved states.
  • Existing methods for solving HMPs often involve complex integral equations or numerical approximations.
  • There is a need for alternative, potentially exact, analytical methods for HMP analysis.

Purpose of the Study:

  • To develop a master equation for HMP distributions.
  • To solve the HMP problem by transforming it into a functional equation.
  • To derive an exact analytical expression for the entropy of HMPs.

Main Methods:

  • Formulation of a master equation for HMP distributions.
  • Solving the master equation via an equivalent functional equation.
  • Analysis of the HMP as a generalized random walk on a 1D strip.

Main Results:

  • An exact mapping between HMP solutions and functional equation solutions was established.
  • An exact expression for the entropy of HMPs was derived, offering an alternative to integral equation methods.
  • The derived solution for two second-order matrices can be generalized for L states and M observables.

Conclusions:

  • The functional equation approach provides an exact and alternative method for solving HMPs.
  • The generalized random walk perspective simplifies the understanding and solution of HMPs.
  • The method is scalable and can be extended to more complex HMP models.