Upgrading weight parameters to random variable in Gaussian mixtures In a Gaussian mixture model we model a density like:
$p(\mathbf{x}|\pi,\mu,\sigma)=\sum \pi_i N(\mathbf{x}|\mu_i,\sigma_i)$ [1]
where $\pi,\mu$ and $\sigma$ are parameters.
I would like to know if the following model is of any use / has a name.
Let's suppose that $\mathbf{\pi} \sim Dirichlet(\mathbf{\alpha})$, so that the weights of the Gaussian become a random variable. Than instead of [1] we have:
$p(\mathbf{x}|\alpha,\mu,\sigma)=\int d\mathbf{\pi}  p(\mathbf{\pi} |\alpha) \sum \pi_i N(\mathbf{x}|\mu_i,\sigma_i)$ [2]
Does it make any sense to use [2] instead of [1] in some setting ? Does this operation make any sense in some setting ?
 A: Trying to answer Xi'an comment, to check if I am understanding his comment. As far as I am understanding he means that upgrading $\pi$ to random variable would have the meaning of modelling our uncertainty of $\pi$ but not the sampling process of the data points. I try to write this better for exercise.
In the standard setting, when we fit a Gaussian mixtures the $\pi$ are parameters. Let's us suppose $\mu$ and $\sigma$ fixed here and suppose we have $M$ samples. In the usual setting the EM algorithm finds the maximum likelyhood estimate:
$\pi_{MLE}=argmax_{\pi} \ log L(\mathbf{x}|\pi)$
Now instead of treating $\pi$ as a parameter we can upgrade it to a random variable. In order to do this we need to define its probability distribution and relation to $x$. In mathematical terms we need:

*

*$P(\mathbf{x}|\pi)$ , that we have ;


*A prior  $P(\pi)$ ;
In order to do this we introduce a parameter $\alpha$ that defines the prior as a Dirchlet distribution and arrive at the graphical model:

which fully defines the joint $P(\mathbf{x},\pi|\alpha)$.
So what did we lose/gain with respect to MLE ?

*

*Still all values of $\pi$ are possible/taken into account. If we condition on the value of  $\pi$ we have $P(\mathbf{x}|\pi,\alpha)=P(\mathbf{x}|\pi)$. This is exactly the parametric likelyhood we had before ;


*Still we can do inference of the value of $\pi$ by estimating the posterior $P(\pi|\mathbf{x},\alpha)$ ;


*As a drawback, we introduced a parameter $\alpha$ which was not present before. Maybe  it would make any sense to estimate $\alpha$ by maximizing the marginal likelyhood:
$$S(\alpha)=\int d\pi P(\pi|\alpha)P(\mathbf{x}|\pi)$$
even if this approach seems to suppose that the uncertainty in $\pi$ is related to the sampling process of $\mathbf{x}$. I have the impression that this way we would mix "uncertainties" and "sampling probabilites", but maybe these concepts are so interrelated that can be mixed together.
Actually this approach ( maximum marginal likelyhood ) in some sense is making the $\pi$ variables disappear the way described in the original message (weird...)
