In Rasmussen's paper it is introduced a Gibbs sampler to make inference about a standard Gaussian Mixture Model.
To simplify, assume the 1-d case with basic hierarchical structure, that is:
$x_i|z_i= k \sim N(\mu_k,\sigma^2_k)$
with mixing weight $\pi_k$ and standard conjugate priors:
$\mu_k \sim N(\mu_0,\sigma^2_0) \quad \quad \sigma^2_k \sim InvGamma(\gamma,\beta) \quad \pi_k \sim Dir(\alpha/k,...,\alpha/k)$
Now taking into account the latent variables $z_i$, in order to implement the Gibbs Sampler I would write first the full conditional distribution of $\mu_k$,$\sigma^2_k$,$\pi_k$,$z_i$, while the paper completely ignore $\pi_k$ and makes inference about the mixing weight through the inference of the indicator variables.
It is somehow related to the Collapsed Gibbs?