A Gaussian mixture model (GMM) represents a probability distribution as a weighted combination of Gaussian components.

For $K$ components,

$$p(x)=\sum_{k=1}^K \pi_k,\mathcal N(x\mid\mu_k,\Sigma_k),$$

where mixture weights satisfy $\pi_k\ge0$ and $\sum_k\pi_k=1$.

Unlike k-means, a GMM gives a soft assignment. For observation $x$, the posterior responsibility of component $k$ is

$$r_k(x)=\frac{\pi_k\mathcal N(x\mid\mu_k,\Sigma_k)}{\sum_j\pi_j\mathcal N(x\mid\mu_j,\Sigma_j)}.$$

If a point lies between two components, both can receive substantial responsibility instead of forcing an all-or-nothing cluster label.

Covariance matrices let components represent ellipsoidal clusters with different orientations and spreads, so GMMs are more flexible than the roughly spherical geometry favored by ordinary k-means.

The parameters and latent component assignments are typically fitted with the expectation-maximization algorithm. As with all finite mixtures, the number of components is a modeling choice and a component need not correspond to a naturally meaningful real-world category.