###Short overview about Expectation minimization :
- Marginal likelihood
Expectation minimization contrasts with 'regular' likelihood minimization by refering to the minimization of a marginal likelihood.
$$\underbrace{p(X\vert \theta)}_{\substack{\text{marginal likelihood}\\\text{ $\mathcal{L}(\theta \vert X)$}}} = \int_z \underbrace{p(X, z \vert \theta)}_{\substack{\text{likelihood}\\\text{ $\mathcal{L}(\theta \vert X,\underset{\uparrow \\ \substack{\llap{\text{This $z$ is }\rlap{\text{missing data}}}}}{z})$}}} \text{d}z = \int_z p(X \vert \theta,z) p(z\vert X,\theta) \text{d}z $$
So this relates to an integral over some likelihood with an additional parameter $z$ (e.g. missing data).
- EM algorithm
In the EM algorithm this integral is not minimized directly:
$$\hat\theta = \underset{\theta}{\text{arg min}} \left( \int_z p(X \vert \theta,z) p(z\vert X,\hat\theta) \text{d}z \right)$$
but instead it is computed in an iterative way:
$$\hat\theta_{k+1} = \underset{\theta}{\text{arg min}} \left( \int_z p(X \vert \theta,z) p(z\vert X,\hat\theta_k) \text{d}z \right)$$
This is done by picking an initial $\theta_1$ and updating repetitively. Note that now the optimization keeps the term $p(z\vert X,\theta)$ fixed.
###Not all problems are like that.
So this marginal likelihood is only the case for problems with unobserved data. For instance in finding a Gaussian Mixture (example on wikipedia) one may consider an observed variable $z$ that refers the class (which component in the mixture) that a measurement belongs to.
There are many problems that do not consider a marginal likelihood and evaluate parameters directly in order to optimize some likelihood function (or some other cost function but not a marginalized/expectation one).