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Area of Science:

  • Information Theory
  • Statistical Signal Processing
  • Mathematical Physics

Background:

  • Dębowski (2009) stated invariance of completion and chain rule for Shannon information measures.
  • Existing proofs require correction, particularly for arbitrary fields.

Purpose of the Study:

  • To provide corrected proofs for the invariance of completion and the chain rule of Shannon information measures.
  • To establish auxiliary approximation results for Shannon information measures.
  • To demonstrate the utility of generalized calculus for Shannon information measures in specific applications.

Main Methods:

  • Development of auxiliary approximation results for Shannon information measures.
  • Rigorous mathematical proof construction based on these auxiliary results.
  • Application of generalized calculus to analyze stationary processes and statistical models.

Main Results:

  • Corrected proofs for the invariance of completion and the chain rule for Shannon information measures.
  • Novel auxiliary approximation results for Shannon information measures with potential independent interest.
  • Demonstration of the applicability of the generalized calculus in ergodic decomposition and natural language modeling.

Conclusions:

  • The corrected proofs enhance the theoretical foundation of information measures for arbitrary fields.
  • The auxiliary results offer new tools for information-theoretic analysis.
  • The generalized calculus provides valuable insights into complex systems like stationary processes and natural language.