Real time example: Estimation for incomplete data Following is from Csiszar and Shields' FnT monograph "Information Theory and Statistics":

The expectation–maximization or EM algorithm is an iterative
  method frequently used in statistics to estimate a distribution supposed
  to belong to a given set $\mathcal{Q}$ of distributions on a finite set $A$ when, instead of a "full" sample $x_1,\dots,x_n \in A^n$ from the unknown distribution, only an "incomplete" sample $y_1,\dots,y_n \in B^n$ is observable. Here $y_i = T(x_i)$ for a known mapping $T\colon A\to B$.

Can someone give a real time example of such a situation? What would be $T$ in such an example?
 A: Consider a coin flipping experiment, where we are given two coins: A and B, with unknown probabilities of flipping Heads: $\theta_a$ and $\theta_b$. Our goal is to predict these $\theta$ values, i.e. a given set $Q$ of distributions, for each coin at the end of our experiment. 
The experiment involves randomly picking either coin A or B, flipping it 10 times and recording all the results. It is depicted as below. Note that we know the identity of our coin in all five random coin picks. Going back to your question, this is the full data-set version.

Now, imagine we do the same thing as above but this time we do not know whether the coin that we randomly pick is A or B. There is a known mapping $T$ between the coin $id \in \{A, B\}$ -> $\theta_{id}$ which we cannot directly observe. So now we have incomplete data. The E-M solution for this is as depicted below. This is explained in the paper by Do and Batzoglou, so I won't go into more detail.

If you want to dig deeper, relevant code is provided in my previous answer here.
As a side note, the coin labels A and B have nothing to do with the $A$ and $B$ you define in your answer. This is potentially confusing, but I kept to them to be consistent with the figures from the paper.
