I am reading a paper where author uses EM for the following equation to find the parameters $\theta$(and $\beta$) : $$ J=\sum_m \alpha_{m}\sum_i\sum_j w_{mij}\log\sum_k \theta_{ik}\beta_{mjk} $$ where: $$ \sum_k\theta_{ik}=1 \ \ \forall i $$ The paper uses the hidden variable z in the following form which I have no problem with that : $$ p(z_{mij}=K|\theta^{t-1},B_m^{t-1})=\frac{\theta_{iK}^{t-1} \beta_{mjK}^{t-1}}{\sum_k \theta_{ik}^{t-1} \beta_{mjk}^{t-1} } $$
But in the M step, I can't figure out how the paper calculates the following formula:
$$ \theta_{ik}^{t}\propto\sum_m \alpha_{m}\sum_jw_{mij}p(z_{mij}=k\mid\theta^{t-1},B_m^{t-1}) $$
It might help to see $\alpha$,$\theta$ and $\beta $ as $1d$, $2d$ and $3d$ arrays.
For more information you can find the paper available here :http://cs.ucsb.edu/~xyan/papers/kdd12_metapath.pdf [section 4.1]