Consider some Gaussian Mixture consisting of a distribution $p$ on $x \in \mathbb{R}^d$ such that:
$$\forall x \in \mathbb{R}^d, p(x) = \sum_{i=1}^n \frac{1}{n} N(x | \mu_i,\Sigma_i) $$
with all $\Sigma_i$ being diagonal. Now, suppose $n$ is very large, and we want to study some statistical properties of $p$, like entropy or divergence with another similar mixture. The fact that $n$ is very large often makes (to my knowledge) the computation of the density ($pdf$) of $p$ untractable, which is a problem for the estimation of the mentionned values.
My question is: is there some known approach to "merge" the multivariate Gaussians, that is approximate $p$ by a smaller Gaussian Mixture ?