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 &\propto \text{Gamma}(a,1) \enspace \\[.5em] \omega &\propto \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, $\beta$ 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 but appears in the Variational EM.