I've been studying the Expectation Maximization algorithm. According to the formula shown here, what I have to do in the M step is to compute a new $\theta$ that maximizes the conditional expectation of the log function, which is $\ln P[X, z|\theta] $: http://i86.photobucket.com/albums/k118/ProtoMan_03/expectation_zps2689ab59.jpg
( The picture above can be acquired in page 8 of this tutorial: http://www.seanborman.com/publications/EM_algorithm.pdf )
However, in the coin toss example below: http://www.nature.com/nbt/journal/v26/n8/full/nbt1406.html?pagewanted=all
$\ln P[X, z|\theta] $ is nowhere to be found, and they don't prove how the new $\theta^{t+1}$ they got after each iteration is better than the $\theta^t$ previously acquired.