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Maximum Entropy Estimation of Probability Distribution of Variables in Higher Dimensions from Lower Dimensional Data.

Jayajit Das1, Sayak Mukherjee2, Susan E Hodge2

  • 1Battelle Center for Mathematical Medicine, Research Institute at the Nationwide Childre's Hospital, 700 Children's Drive, OH 43205, USA; Department of Pediatrics, The Ohio State University, Columbus, OH 43205, USA; Department of Physics, The Ohio State University, Columbus, OH 43210, USA; Department of Biophysics Program, The Ohio State University, Columbus, OH 43210, USA.

Entropy (Basel, Switzerland)
|February 5, 2016
PubMed
Summary

This study introduces a new maximum entropy (MaxEnt) method to infer unknown probability distributions when the relationship between variables is non-unique. The approach estimates distributions using available data without needing explicit Lagrange multipliers.

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Area of Science:

  • Statistical inference
  • Probability distributions
  • Information theory

Background:

  • Inferring an unknown distribution Q(x) from a known distribution P(y) is common.
  • Typically, the dimension of X (n) is less than or equal to the dimension of Y (m), simplifying the inference.
  • However, cases where n > m present challenges due to non-unique functional relationships.

Purpose of the Study:

  • To propose a novel maximum entropy (MaxEnt) approach for estimating unknown distributions Q(x) when n > m.
  • To provide a method that infers Q(x) solely from the known distribution P(y) without explicit Lagrange multipliers.
  • To develop and validate this MaxEnt approach for both discrete and continuous probability distributions.

Main Methods:

  • Developed a maximum entropy (MaxEnt) framework for distribution inference.
  • Applied the method to scenarios with non-unique functional relationships (n > m).
  • Validated the approach for discrete and continuous probability distributions.

Main Results:

  • The proposed MaxEnt method effectively estimates unknown distributions Q(x) even when the relationship Y→X is non-unique.
  • The method successfully infers Q(x) using only the known distribution P(y).
  • The approach avoids the need for explicit calculation of Lagrange multipliers.

Conclusions:

  • The novel MaxEnt approach offers a robust solution for inferring probability distributions in complex scenarios (n > m).
  • This method provides valuable inferences about Q(x) in the absence of complete information.
  • The approach is validated and demonstrated with illustrative examples.