I've recently stated reading about mixture models and variational inference in this excellent paper, but I'm having troubles dissecting the models described, and have a couple of questions. Please see picture provided:
So the point here is to find out which latent variables are most probable given our observations?
What I'm having trouble interpreting is equation 7 and 8. Can I regard equation 7 as the numerator in equation 2, where mu and c are z (the latent variables?), and equation 8 as the denominator in equation 2?
I'm also having troubles understanding how they've derived equation 7 and 8. For equation 7, Is it some chain rule of the joint? And in equation 8, are they simply just marginalizing out each xi, and taking the product of each sample assuming they're i.i.d?
Since my calculus isn't really up to par, I assume the key here is that they want to show that integral in equation 8 is intractable, thus the need for Variational Inference?
I also have a question regarding the hyper parameter for the Gaussian for the means, we assume it is set (to something) in this model right? And how would I choose this if I were to create a model on my own?