I just start working with Gaussian mixture models and I just confused about some information, which I really would like to make sure that I understand the model very well.
A Gaussian mixture model is a weighted sum of $m$-components Gaussian densities. Hence, each component is assumed to be Gaussian.
My questions are:
1- As I understand the mixture components are used to describe the dependencies structures. So, do these components have any information about the univariate margin of each variable?
In other words, the joint distribution function includes the information of the margin and the dependencies structures. Unlike the copula models, we cannot decouple the margins from the dependencies structures in, for example, Gaussian mixture models. So, the joint distribution includes all the information about margins and the dependency among variables. So, if the mixture components describe the dependencies structures, from where can I get the information of the margins?
By margin, I mean the distribution of each variable.
2- Gaussian mixture models can describe any type of distributions and that may require a very large number of mixture components. Is that correct? or there is no relationship between the type of the mixture dependencies and the number of mixture components? For example, if I have very complex mixture dependencies. From the scatter plot, I find the dependence takes circle shape. Hence, I will need to, say, 20 mixture components to describe this type of dependency.