[Math] Joint Distribution of Two Gaussian Mixture Model

Question

I have two GMM distribution (K=2) of multi-variant Gaussian distributions whose parameters(mean and sigma vectors and weight)are found using the EM algorithm. How can I find the the joint distribution of the two GMMs?

Answer 1

Suppose that the density of 2 GMMs are $f_1(x)=\pi_1\phi_{\theta_{1,1}}(x) + (1-\pi_1) \phi_{\theta_{1,2}}(x)$ and $f_2(x)=\pi_2\phi_{\theta_{2,1}}(x) + (1-\pi_2) \phi_{\theta_{2,2}}(x)$ respectively.

Let $p$ be the prior probability of $x$ generated by GMM1. Then the joint density is $$f(x)=p*f_1(x)+(1-p)*f_2(x)$$ which is the density of a GMM with $K = 4$ since $$p\pi_1 + p(1-\pi_1) + (1-p)\pi_2 + (1-p)(1-\pi_2)=1$$