Assumption of Gaussian mixture model I am still confused when I read about the Gaussian Mixture Model and how does it work.
One thing that I do not understand is that "GMM assumes the data is a mixture of Gaussian or
i.i.d (Independent and identically distributed) to model their distribution". Does this mean GMM can't deal with non-Gaussian distribution or non-i.i.d data as this violates its assumption?
 A: The sentence on i.i.d. variables says that for all $i$
$$
X_i \overset{i.i.d.}\sim f
$$
This means that $X_i$'s are independent and each of them follows the distribution $f$. In the case of Gaussian mixture, $f$ is the Gaussian mixture distribution.
This doesn't mean that $X_i$'s are Gaussian, they come from a Gaussian mixture. So the data can be non-Gaussian as long as it can be modeled using a Gaussian mixture. The Gaussian mixture does not have to be bell-curved or anything like that, it can have all kinds of shapes, depending on its components and their mixing weights.
To give an example, say that around the corner you have an alcohol mixture shop. You pay a fixed amount and with probability $\pi$ you get a random beer and with probability $1-\pi$ you get a random wine. The shop does not sell only beer or only wine, and when you make a purchase you don't know if it will be a beer or wine. It won't be anything else, but all the products come from the same categories and not from other categories.
