# Variational inference in the directed network in Airoldi's Mixed Membeship Stochastic Blockmodel

$\newcommand{\E}{\mathbb{E}}$I was dealing with derivations for the variational parameters in the model proposed by Airoldi, Blei, Fienberg, and Xing, 2008 for overlapping communities using a mixed-membership-stochastic-blockmodel, and I was a bit confused in terms of treating the $z_{p\rightarrow q}$ and $z_{q\rightarrow p}$.

I know exactly what they mean although when dealing with the derivations in the appendix I see a shift in notation from the lower bound/evidence lower bound (ELBO) in the appendix B.2.

I don't exactly know how: $$\sum_{m,p,q,g} \phi^m_{p\leftarrow q,g}\E[\log \pi_{q,k}]$$ turns into: $$\sum_{m,p,q,g} \phi^m_{p\leftarrow q,g}\E[\log \pi_{p,k}],$$ which indeed would result in the parameter $\gamma_{p,k}$ estimated as: $$\alpha_k + \sum_{m,q}\phi^m_{p\rightarrow q,k} + \sum_{m,q}\phi^m_{p\leftarrow q,k}.$$

What I get is: $$\alpha_k + \sum_{m,q}\phi^m_{p\rightarrow q,k} + \sum_{m,q}\phi^m_{q\leftarrow p,k},$$ if I do some rearrangements.

Can anybody help me where I am going wrong?

• I think that is just a typo in the paper, and the correct form is $\alpha_k + \sum_{m,q}\phi^m_{p\rightarrow q,k} + \sum_{m,q}\phi^m_{q\leftarrow p,k}$. – A.Yazdiha Oct 26 '16 at 8:10
• Which formulas are confusing you? I don't see a $\theta$ term in B.2 of this version of the paper. – Sean Easter Nov 14 '16 at 21:49
• that is indeed the $\pi$ in the paper – A.Yazdiha Nov 15 '16 at 9:32