What is the problem with this model parameter estimation algorithm?

Question

In a statistical model with parameters $\theta$ and unobserved laten variables $Z$, the model likelihood is

$$L(\theta;X)=Pr(X|\theta)=\sum_ZPr(X,Z|\theta)$$

The standard way to estimate $\theta$ is to maximize this likelihood (e.g. using EM or so). However if this estimation is computational intractable, can I maximize $Pr(X,Z|\theta)$ w.r.t both $Z$ and $\theta$, and use this estimated $\theta$ as the model parameter? Does this method have a name? What is the main drawback or problem with this method?

Answer 1

The problem with that approach is that every time you add an observation of $X$, you also add an "observation" of $Z$. The space in which you want to find the optimal values of $Z$ and $\theta$, thus grows very large. Instead of making the problem easier, I think you make it more complicated by adding $Z$.

If the optimization problem is intractable, you might consider using a smaller (or simpler) $\theta$. Your question does not give any details about the distribution of $X$ or about $\theta$, therefore I cannot give more concrete solutions.