The integral form of student's t distribution is given as follows [1]:
$p(x|\mu,\lambda,\nu)=\int_0^\infty \textrm{Normal}(x|\mu,(\lambda\eta)^{-1})\textrm{Gamma}(\eta|\nu/2,\nu/2)d\eta~~~~~~~~~~~~$ (1)
The complete data likelihood is written as follows:
$P(X,Z|\theta)=\prod_{i=0}^N\textrm{Normal}(x_i|\mu,(\lambda\eta_i)^{-1})\textrm{Gamma}(\eta_i|\nu/2,\nu/2)~~~~~~~~~~$(2)
I would like to ask why $\eta$ becomes $\eta_i$ when we're writing the likelihood. There are $N$ observations, single $\mu$, and $N$ instances of $\eta$. How can you see this in (1)? Is there a 1-to-1 relation between $x_i$ and $\eta_i$, or is it just a coincidence that there are $N$ instances of both? Lastly, can we see Gamma() here as a prior distribution over $\eta$ since when doing EM we find the posterior latent distribution $p(Z|X,\theta)$?