I have some questions concerning estimating maximum likelihood of Gaussian mixture model.
- As I have read around some material, they usually use EM algorithm for maximizing the complete likelihood function (with hidden variables). If without introducing hidden variables, can I optimize directly the incomplete likelihood function? That is I consider $\Theta = \left( \pi, \theta \right)$ are the mixing proportion vector and the parameters vector, respectively, and then I compute the gradient of likelihood $\nabla L(\Theta)$ and run gradient methods.
- What is the benefits of introducing hidden variables and is there any reasons people tend to use EM over gradient methods?