Suppose we would like maximize a likelihood function $p(\mathbf x, \mathbf z| \theta)$, where $\mathbf x$ is observed, $\mathbf z$ is a latent variable, and $\theta$ is the collection of model parameters. We would like to use expectation maximization for this.
If I understand it correctly, we optimize the marginal likelihood $p(\mathbf x|\theta)$ as $\mathbf z$ is unobserved. However, this is counterintuitive to me.
If $\mathbf z$ is unobserved, I think of it as another model parameter. Therefore, for maximum likelihood estimation, we should find $\mathbf z, \theta$ such that $p(\mathbf x|\mathbf z, \theta)$ is maximized.
So, my question is why is it standard to optimize $p(\mathbf x|\theta)$ instead of $p(\mathbf x|\mathbf z, \theta)$?
I have searched through several explanations of EM, but could not find answer to this question.