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A Maximum Entropy Procedure to Solve Likelihood Equations.

Antonio Calcagnì1, Livio Finos1, Gianmarco Altoé1

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Summary
This summary is machine-generated.

We introduce a novel maximum entropy (ME) approach to solve likelihood equations. This method reformulates score parameters, offering a viable alternative to standard procedures, especially in challenging logistic regression scenarios.

Keywords:
binary regressiondata separationmaximum entropymaximum likelihoodscore function

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Area of Science:

  • Statistics
  • Information Theory

Background:

  • Standard methods for solving likelihood equations involve setting the score function to zero.
  • Maximum likelihood estimation can be challenging in specific cases, such as logistic regression with data separation.

Purpose of the Study:

  • To propose and evaluate a maximum entropy (ME) approach as an alternative method for solving likelihood equations.
  • To assess the efficacy of the ME approach, particularly in scenarios where standard methods face difficulties.

Main Methods:

  • Reformulating score parameters as expected values of discrete probability distributions.
  • Maximizing Shannon's entropy function with the score function as an informative constraint.
  • Empirical case studies and a simulation study, including logistic regression under data separation.

Main Results:

  • The maximum entropy reformulation successfully solves the likelihood equation problem.
  • The ME approach yields results comparable to Firth's bias-corrected method in difficult logistic regression cases.
  • Parameter search is simplified by reparameterization into a smaller (hyper) simplex space.

Conclusions:

  • The maximum entropy approach offers a viable alternative for solving likelihood equations.
  • This ME method demonstrates effectiveness even when maximum likelihood estimation is problematic.
  • The findings suggest ME as a valuable technique in statistical inference.