Suppose I have a Dirichlet Process Mixture model defined as follows:
$\alpha \sim G(a,b)\\ \pi|\alpha \sim \text{Dir}(\alpha)\\ z|\pi \sim \text{Cat}(\pi)\\ $
where $G$ is just a standard Gamma distribution, and
$ \mu,\Sigma \sim NIW(\beta) $ where $NIW$ stand for Normal Inverse Wishart.
Ultimately the plate model looks like:
Upon doing some probabilistic maniuplation I arrive at the following terms I would like some clarification on: $p(z|X),p(z|X,\alpha),p(X|z)$. I am just wondering how to interpret these.
Is $p(z|X,\alpha) = p(z|\alpha)$? It seems that $X$ is already implicitly used in the $p(z|\alpha)$ expression as the `number of terms allocated to a cluster' in the sense of the Chinese Restaurant Process, when considering $p(z|\alpha)$ by itself, so does conditioning on $X$ change anything?
How should I interpret the expressions $p(z|X)$ and $p(X|z)$? It feels strange because $z$ just acts like a `switch' or index for this Bayes Net, so interpreting either direction of these feels strange.
In addition, if I add a (Gamma) prior distribution over the $\alpha$ term, with a likelihood term similar to that suggested by Escobar & West 1995 (equations 13 / 14) then is it possible to make the claim that this empirical distribution over $\alpha$ has a regularising effect over the amount of clusters forming (or in the cluster allocations for each point, that is $p(z_i = k | X, \beta, \alpha, z_{\lnot i}$).