Maximum Likelihood Estimates for Gaussian Mixtures Are Transcendental

  • Carlos AméndolaEmail author
  • Mathias Drton
  • Bernd Sturmfels
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9582)


Gaussian mixture models are central to classical statistics, widely used in the information sciences, and have a rich mathematical structure. We examine their maximum likelihood estimates through the lens of algebraic statistics. The MLE is not an algebraic function of the data, so there is no notion of ML degree for these models. The critical points of the likelihood function are transcendental, and there is no bound on their number, even for mixtures of two univariate Gaussians.


Algebraic statistics Expectation maximization Maximum likelihood Mixture model Normal distribution Transcendence theory 



CA and BS were supported by the Einstein Foundation Berlin. MD and BS also thank the US National Science Foundation (DMS-1305154 and DMS-1419018).


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Carlos Améndola
    • 1
    Email author
  • Mathias Drton
    • 2
  • Bernd Sturmfels
    • 1
    • 3
  1. 1.Technische UniversitätBerlinGermany
  2. 2.University of WashingtonSeattleUSA
  3. 3.University of CaliforniaBerkeleyUSA

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