# Maximum Likelihood Estimates for Gaussian Mixtures Are Transcendental

arXiv: https://arxiv.org/abs/1508.06958 · HTML: https://mlsanity.com/paper/1508.06958/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Aug 2015.

## Built on by (most cited first, 3 of 3)

- [Fast approximations of the Jeffreys divergence between univariate Gaussian mixture models via exponential polynomial densities](https://mlsanity.com/paper/2107.05901/) (2021)
- [An Efficient 1 Iteration Learning Algorithm for Gaussian Mixture Model And Gaussian Mixture Embedding For Neural Network](https://mlsanity.com/paper/2308.09444/) (2023)
- [Diffusion Model Conditioning on Gaussian Mixture Model and Negative Gaussian Mixture Gradient](https://mlsanity.com/paper/2401.11261/) (2024)

## Abstract (arXiv)

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.