I have problem distinguishing between the latent variables $z_i$ and the parameters $\theta_i$ in EM algorithm. Suppose we have the hierarchical priors

\begin{aligned}
\beta|\tau,\omega &\sim \mathcal{N}(0,\tau\,\omega) \\[.5em]
\tau &\sim \text{Gamma}(a,1) \enspace \\[.5em]
\omega &\sim \text{Inv-Gamma}(b,1) \enspace .
\end{aligned}

In a paper I have read, the latent variables $z_i$ are chosen to be $\{\beta,\tau,\omega\}$ while the hyperparameters $\theta_i$ are $\{a,b\}$.

However, [in other models][1], $\beta$ is chosen as a member of $\theta_i$.

My question is how do we choose $\theta_i$ and $z_i$? Are we free to choose?

Also, why is it that $\theta$ doesn't appear in [Variational Inference][2] but appears in the [Variational EM][3]. 


  [1]: https://stats.stackexchange.com/questions/134207/question-on-how-to-use-em-to-estimate-parameters-of-this-model?rq=1
  [2]: https://fabiandablander.com/r/Variational-Inference.html
  [3]: https://chrischoy.github.io/research/Expectation-Maximization-and-Variational-Inference/