From the book "[Machine Learning a probabilistic Perspective][1]", 
I'm reading about finite/infinite mixture models. Particularly at paragraph 25.2.1 it's stated: 

>  The usual representation (of a finite mixture model) is as follows: 
> 
> $p(x_i|z_i = k, \boldsymbol\theta) = p(x_i|\boldsymbol\theta_k)$
> 
> $p(z_i = k| \boldsymbol\pi = \pi_k) = \pi_k$
> 
> $p(\boldsymbol\pi|\alpha) =\text{Dir}(\boldsymbol\pi|(\alpha/K)\boldsymbol1_K)$ 
> 
> The form of $p(\boldsymbol\theta_k|\lambda)$ is chosen to be be
> conjugate to $p(x_i|\boldsymbol\theta_k)$.  We can write
> $p(x_i|\boldsymbol\theta_k)$ as $\boldsymbol{x}_i \sim F(\boldsymbol\theta_{z_i})$ where F is the observation distribution.
> Similarly, we can write $\boldsymbol\theta_k \sim H(\lambda)$, where H
> is the prior.

Now this modelling is quite confusing to me. What is the difference between $\boldsymbol\theta_k$ and $\boldsymbol\theta_{z_i}$? What is meant by "Observation distribution"? Can we apply EM algorithm to this model, how?  


  [1]: https://amzn.to/2SL8qvh