What is a mixture of finite mixtures? A mixture of finite mixture models seem to be an interesting Bayesian (?) approach to solving clustering with an unknown $k$ number of components. It seems though, unlike the mixture model with a Dirichlet process prior, using a symmetric Dirichlet prior means $k$ doesn't necessarily grow with the size of the dataset $n$. 
Unfortunately, I don't really understand how they are defined, or much else about them. The definition I am working with is from the paper by Miller and Harrison (2017) is as follows:

We consider the following well-known model:

$$
K ∼ p_K, \text{where }p_K \text{ is a p.m.f. on } \{1, 2, \ldots \} \\
(\pi_1, \ldots , \pi_k) ∼ \mathrm{Dirichlet}_k(γ, \ldots , γ), \text{ given }K = k \\
Z_1, \ldots , Z_n ~ \pi, \text{ given } \pi  \\
\theta_1, \ldots , \theta_k ∼ H, \text{ given } K = k\\
X_j ∼ f_{θ_{Z_j}} \text{ independently for }j = 1, \ldots , n, \text { given } θ_{1:K}, Z_{1:n}.
$$

Here, $H$ is a prior or “base measure” on $\Theta ⊂ \mathbb{R}^\ell$,
  and ${f_\theta : \theta \in \Theta}$ is a family of probability
  densities with respect to a sigma-finite measure $\zeta$ on $\chi \subset \mathbb{R}^d$. (As usual, we give Θ and $\chi$ the Borel
  sigma-algebra, and assume $(x, \theta) \rightarrow f_\theta(x)$ is
  measurable.) We denote $x_{1:n} = (x_1, \ldots , x_n)$. Typically,
  $X_1, . . . , X_n$ would be observed, and all other variables would be
  hidden/latent. We refer to this as a mixture of finite mixtures (MFM)
  model.

I think I understand the first three lines of definitions here but the rest seems very general and opaque. 
What is finite mixture of mixtures? What is the intuition here? If people will humour me, I wouldn't mind also knowing :


*

*Why is there all this measure-theoretic terminology in the definition?

*How do I fit such a model? What is reversible jump Markov chain Monte Carlo and why is it so strongly associated with this model?

*Why isn't this more commonly used as it seems to be a nice solution to the problem of clustering with unknown $k$? 



Mixture models with a prior on the number of components, J. W. Miller
  and M. T. Harrison, Journal of the American Statistical Association
  (JASA), Vol. 0, 2017, pp. 1-17.

 A: Here is my understanding,
we draw the index or indicator as follows to indicate which $\theta_j$ to use
$$
Z_i \sim \text{Categorical}(\pi_1, ..., \pi_k), i=1,...,n
$$
then we draw the parameters of each component of a mixture model as,
$$
\theta_j \sim H, j = 1,...,K
$$
then we draw the sample according to the selected parameter $\theta_{Z_i}$
A: I started working on Mixtures of Finite Mixture this week, and my understanding is the following, (it might be wrong but seems to make sense). I started with the paper Generalized Mixtures of Finite Mixtures and
Telescoping Sampling.
A simple Finite Mixture model is a model defined as
$$p_{K}(y) = \sum_{i=1}^{K}\eta_{k}f(y|\theta_{k}) \ \ (1)$$
equivalently an observation $y$ comes from the mixture component $f(y|\theta_{k})$ with probability $\eta_{k}.$ So, (1) is a mixture of densities with different mixture components $\theta_{k}$
Now since we call a Mixture of Finite Mixtures, we want a mixture of such things that we defined as (1). The (1) is conditional on the number of mixture components $K$, so a natural way to create a Mixture of Finite Mixtures, is if you let the mixing happen on $K$, and you can do that by letting $K$ being random and unknown. So, you can have something like
$$p(y) = \sum_{K=1}^{\infty}p(K)p_{K}(y)$$
So, equivalently you can say an observation $y$ comes from the Finite Mixture $p_{K}(y)$ with mixture probability $p(K)$ (can be regarded as the prior distribution over $K$). So, you can see that you have some nested mixture models, so I guess that comes from where the names Mixture of Finite Mixtures come.
