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This summary is machine-generated.

We introduce the Hyperbolic Sum Rule (HSR) for calculating probabilities of dependent events, proving it

Keywords:
AIBayes’ TheoremDSPMLQGTVenn diagramentropy

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Area of Science:

  • Probability theory
  • Information theory
  • Statistical mechanics

Background:

  • The Conventional Sum Rule (CSR) excludes recursive probabilities, limiting its applicability.
  • Recursive dependencies in events are common in AI and machine learning.
  • Current methods for handling complex probability calculations are often intractable.

Purpose of the Study:

  • Introduce and validate the Hyperbolic Sum Rule (HSR) for probabilities.
  • Demonstrate the Maximum Entropy (MaxEnt) property of the HSR.
  • Establish the physical nature of probability and its connection to thermodynamics.

Main Methods:

  • Derivation of the HSR, showing its isomorphism to the hyperbolic tangent double-angle formula.
  • Proof of the HSR's Maximum Entropy (MaxEnt) property.
  • Analysis of recursive dependencies and their implications for probability calculations.

Main Results:

  • The HSR accurately calculates probabilities for recursively dependent events.
  • HSR is proven to be Maximum Entropy (MaxEnt).
  • HSR enables scalable and analytical calculations, unlike the CSR.

Conclusions:

  • The HSR should be the default method for probability calculations due to its broader applicability.
  • Probability is a physical quantity, not merely a mathematical construct.
  • The HSR has implications for digital signal processing and Quantitative Geometrical Thermodynamics.