Hi,
Thank you for implementing such a great method! I have a question regarding the objective function used for MAP estimation. According to the paper, the objective is: $E_{q(c)q_{\phi}(\mu | u)}[\log p(z,c,\pi, \mu, \Lambda,\alpha)]$. However, an alternative objective function that is more natural to me is to maximize $\log E_{q(c)q_{\phi}(\mu | u)} p(z,c,\pi, \mu, \Lambda,\alpha)$, and then we can marginalize out both $c$ and $\mu$ to have an explicit form of $\log p(z,\pi, u, \Lambda,\alpha)$, which can be directly optimized rather than using EM framework. Is there any reason that you decided to pick $E_{q(c)q_{\phi}(\mu | u)}[\log p(z,c,\pi, \mu, \Lambda,\alpha)]$ instead of $\log E_{q(c)q_{\phi}(\mu | u)} p(z,c,\pi, \mu, \Lambda,\alpha)$?
In addition, since $p(z|c=j)$ in VMM is modeled as $\mathbb{E}[\mathcal{N}(z|\mu_j,\Lambda_{j}^{-1})]$, I think we should have $q(c|z)=softmax(\log(p(c)p(z|c)))=softmax(\log\pi + \log \mathbb{E}[\mathcal{N}(z|\mu,\Lambda^{-1})])$ instead of $softmax(\log\pi +\mathbb{E}[ \log \mathcal{N}(z|\mu,\Lambda^{-1})])$ shown in Appendix B, do you have any reason on why you chose the latter form?
Thanks!
