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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, $\beta$ 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.

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    $\begingroup$ I worry that there are a number of misunderstandings here. Where do you ever see that "𝛽 is chosen as a member of 𝜃𝑖" or that VI is performed without parameters? $\endgroup$
    – Arya McCarthy
    Commented Jul 7, 2022 at 23:35
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    $\begingroup$ @AryaMcCarthy, if you check the answer in stats.stackexchange.com/questions/134207/… $\beta$ is chosen as $\theta_i$ and $w_i$ as latent variable. $\endgroup$
    – Isaac
    Commented Jul 7, 2022 at 23:49
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    $\begingroup$ @AryaMcCarthy, here fabiandablander.com/r/Variational-Inference.html only $z_i$ appears without $\theta$. $\endgroup$
    – Isaac
    Commented Jul 7, 2022 at 23:55
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    $\begingroup$ @AryaMcCarthy, I am not sure what you mean, could you explain more please. $\endgroup$
    – Isaac
    Commented Jul 8, 2022 at 5:06
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    $\begingroup$ @AryaMcCarthy, The two models look the same....no? $\endgroup$
    – user3911153
    Commented Jul 8, 2022 at 8:31

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