# Why marginal likelihood is optimized in expectation maximization?

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.

If you don't know $$z$$ you cannot condition on $$z$$ by $$p(x|z,\theta)$$, but we can “hallucinate” it for the lower bound function using the parameter we get in the previous step.
Because of the missing data problem. $$z$$ is not observed and missing in our training data.
Ultimately we are optimizing $$p(x|\theta)$$ but it can lead to multiple local maxima and no closed-form solution then we can make it a sequence of subproblems that can be optimized in each step and guaranteed to converge to a local optimum(may be global optimum) by introducing $$q$$.