Say I observe $N$ observations $\{x_1, \dots, x_N\}$ from a $k$ component Gaussian Mixture model, with $k > 0$ known and is such that each $x_i|\boldsymbol{\pi}, \boldsymbol{\mu} \sim \sum_{j=1}^{k} \pi_j \mathcal{N}(\mu_j, \sigma_j)$, with each $\sigma_j$ also known for component $j = 1, \dots, k$. The vector of mixing weights $\boldsymbol{\pi} = (\pi_1, \dots, \pi_k)$ and means $\boldsymbol{\mu} = (\mu_1, \dots, \mu_k)$ are unknown.
Let's say also that the label of each observation to its group is unknown; i.e. let $z_i \in \{1, \dots k\}$ be an allocation label, allocating one of the $k$ groups to observation $i = 1, \dots, N$. Marginally, we have that $\mathbb{P}(z_i = j) = \pi_j$ for $j = 1, \dots, k$. However, I have another unknown parameter $\gamma >0$ which is only related to the number of counts observed from each component; i.e. I know the probability distribution $\mathbb{P}(s_j|\gamma)$, where $s_j = \#(l: z_l = j)$, for each $j$.
I can construct a Gibbs sampler to sample from the conditionals $\boldsymbol{\pi}, \boldsymbol{\mu}$, when their prior distributions are dirichlet and Gaussian respectively. However, I am stuck on finding the conditional distributions of $z_1, \dots z_N$ and $\gamma$ given all other unknown parameters.
Is it true that \begin{align*} f(\gamma| z_1, \dots, z_N) & \propto \mathbb{P}(s_1, \dots, s_k| \gamma) f(\gamma) \\ & \propto f(\gamma)\Pi_j \mathbb{P}(s_j|\gamma), \end{align*} where $f(\gamma)$ denotes the prior distribution of $\gamma$?
And if so, does this mean that to sample from $z_1, \dots, z_N$, that for each $j$, $$ \mathbb{P}(z_i = j|\gamma, z_1, \dots, z_{i-1}, z_{i+1}, \dots, z_N,\boldsymbol{\mu}, \boldsymbol{\pi}) \propto $$ $$\pi_j \exp\left(-\frac{1}{2\sigma_j^2} (x_i-\mu_j)^2 \right) \frac{\mathbb{P}(s_j = d+1|\gamma)}{\int_{0}^{\infty} \mathbb{P}(s_j = d+1|\gamma) f(\gamma) d \gamma},$$ where $d = \#(l \neq i: z_l = j)?$
Or should I further condition on the number of counts $s_j$, for $j=1, \dots,k$? Any help would be kindly appreciated! Thanks.
