If we have complete data $\left\{X, Z\right\}$, it is easy to maximize the likelihood $p(X, Z | \theta)$. Unfortunately, with incomplete data ($X$ only), we must marginalize over the latent variables $Z$, so:

$\ln(p(X|\theta)) = \ln\left(\sum_Z p(X,Z|\theta)\right)$

The parameters $\theta$ inside the logarithm make it difficult to compute the maximum likelihood directly.