# How does the EM algorithm work in Bayesian regression? [closed]

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.

• 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? Jul 7 at 23:35
• @AryaMcCarthy, if you check the answer in stats.stackexchange.com/questions/134207/… $\beta$ is chosen as $\theta_i$ and $w_i$ as latent variable. Jul 7 at 23:49
• @AryaMcCarthy, here fabiandablander.com/r/Variational-Inference.html only $z_i$ appears without $\theta$. Jul 7 at 23:55
• @AryaMcCarthy, I am not sure what you mean, could you explain more please. Jul 8 at 5:06
• @AryaMcCarthy, The two models look the same....no? Jul 8 at 8:31