# GMM with Dirichlet prior

I'm learning about the variational inference - mean field approximation on from this online course DeepBayes2019 page 30

The probabilistic model is written as follows: $$p(X, Z \mid \pi, \mu, \lambda) = Dir(\pi \mid \alpha)\prod_{i=1}^{N} \prod_{k=1}^{K} [\pi_k \mathcal{N}(x_i \mid \mu_k, \lambda_k^{-1})]^{z_{ik}}$$

The E-step is defined as $$p(Z, \pi \mid X, \mu, \lambda) \approx q(Z, \pi) = q(Z) q(\pi)$$ with

$$q(Z) = \prod_{i=1}^{N}q(z_i) \quad q(z_i) = \prod_{k=1}^{K} \rho_{ik}^{z_{ik}} / \sum_{k=1}^{K} \rho_{ik} \\ q(\pi) = Dir(\pi \mid \alpha'), \quad \alpha_k' = \alpha_k + \sum_{i=1}^{N} \mathbb{E}_{q(Z)}z_{ik}$$

And the M-step is: $$\mu_k = \frac{\sum_{i=1}^{N} \mathbb{E}_{q(Z)}z_{ik}x_i}{\sum_{i=1}^{N} \mathbb{E}_{q(Z)}z_{ik}}, \quad \lambda_k = \frac{\sum_{i=1}^{N}\mathbb{E}_{q(Z)}z_{ik}}{\sum_{i=1}^{N}\mathbb{E}_{q(Z)}z_{ik}(x_i - \mu_k)^2}$$

1. My problem is that I don't understand how they defined the $$\rho$$ in the proof they wrote: $$log q(Z)= \sum_{i=1}^{N} \sum_{k=1}^{K} z_{ik}(\mathbb{E}_{q(\pi)} log \pi_k + log\mathcal{N}(x_i \mid \mu_k, \lambda_k^{-1}))+ cst \\ = \sum_{i=1}^{N} \sum_{k=1}^{K} z_{ik} log \rho_{ik} + cst$$

2. I am not sure that I understand the role of the $$z_{ik}$$ matrix