Why Gaussian mixture model uses Expectation maximization instead of Gradient descent? Why Gaussian mixture model uses Expectation maximization instead of Gradient descent?
What other models uses Expectation maximization to find best optimal parameters instead of using gradient descent?
Not all the parameters (e.g., the assignment parameters) for a Gaussian mixture model are smoothly differentiable, thus can not be fit with gradient descent.
Other use cases for the expectation–maximization (EM) algorithm are:

Clustering
Latent variable estimation
Missing data estimation