A Gaussian mixture model is a mixture distribution whose component distributions are normal distributions. Its probability density function has the form
 |
where 
, 
, and 
is the probability
density function of a multivariate
normal distribution with mean 
and covariance matrix 
.
The component label is an unobserved categorical random variable. Model parameters are often estimated by alternately estimating component memberships and maximizing the resulting expected log likelihood function. A Gaussian mixture can be unimodal or multimodal, depending on its weights, means, and covariance matrices.
Okahara and Kobayashi (2026) proved that a mixture of three normal distributions with positive weights and a common positive
definite covariance matrix has at most four
modes in every dimension 
, and that four is attained. If the whitened means
are collinear or form a right triangle or obtuse
triangle, the sharp maximum is three. The result includes degenerate modes. Okahara
and Kobayashi (2026) report AI assistance with symbolic-check design and algebraic
verification code.
