# Does the EM algorithm for mixtures still address the missing data issue?

There is a PDF $p(D| \theta)=p(X,Z| \theta)$ with observed values $X$ but also some missing or incomplete values $Z$ (for eg. resulting from censoring).

The expectation-maximization (EM) algorithm is said to give the parameters $\hat{\theta}$ that maximize the likelihood of $D$--i.e. for both the observed and missing values. Call this "property 1."

The EM algorithm is also used to estimate the weights $w_k$ and parameters $\theta_k$ of a mixture $M(x)$ of $K$ component PDFs $p_c(x| \theta_k)$:

$$M(x)=\sum_{k=1}^{K} w_{k} p_c(x| \theta_k)$$

In this case, the probability that an observation $x_i$ is generated by the PDF component $k$ is treated as the missing values $Z$.

My question is does using the EM algorithm to solve for the weights and parameters of the mixture $M$ that maximize the log likelihood also have "property 1?" I.e. does it maximize likelihood for the whole distribution of both observed and missing values?

To formulate my question I am drawing heavily on this document:

https://courses.csail.mit.edu/6.867/wiki/images/b/b5/Em_tutorial.pdf

Let $X$ be the observed data and $Z$ be the latent data. The EM algorithm is an algorithm to maximize $p(X \mid \theta)$. When you say
The expectation-maximization (EM) algorithm is said to give the parameters $\hat \theta$ that maximize the likelihood of $D$, i.e. for both the observed and missing values. Call this "property 1."
it sounds to me like property 1 is saying the EM algorithm maximizes the likeihood of $p(X, Z \mid \theta)$, as a function of $\theta$. This is impossible - we don't know $Z$, hence there is no way to compute $\hat \theta(X, Z) = \arg\max_\theta p(X, Z \mid \theta)$. What the EM algorithm does is (attempt to) calculate $\hat \theta(X) = \arg \max_\theta \int p(X, Z \mid \theta) \ dZ$, but there is no reason to expect that $\hat \theta(X) = \hat \theta(X, Z)$.