# Use of expectation propagation for inference

I have a PGM as described by the attached diagram.

$y$ is observed and I want to infer the joint posterior distribution as given by $P(w,\lambda, \phi |y)$. Distribution on $w$ is modelled using a multivariate 0 mean with a covariance structure. $\lambda$ needs to be inferred from the data and the prior on it is modelled using a gamma distribution with scale and shape parameters $s_0$ and $C_0$. The noise variance $\phi$ also needs to be estimated and its prior is also modelled using a Gamma distribution with parameters $a_0$ and $b_0$

I have been looking at using Expectation Propagation to perform this approximate inference on this model. As I understand, EP works by minimising the KL divergence between the true posterior and the approximated distributions.

So, my questions are:

1: Can I use EP in such a case?

2: Can someone help me understand what objective function I need to design in this model? As far as I can tell, the EP will approximate the try posterior by independent distributions from the exponential family. So,

$P(w,\lambda, \phi) \approx q(w) q(\lambda) q(\phi)$

And now do we minimise the KL divergence between true p and q's? I am really having trouble understanding this setup and would be really grateful if someone can help me understand where and how to start!