TITLE: KL divergence and mixture of Gaussians
QUESTION [11 upvotes]: Do we have an exact formula to compute the KL divergence between 2 mixtures of Gaussians (i.e convex combinations of a finite number of Gaussian distributions)? 
If not exactly known, are there good upperbounds that are known for this quantity?

REPLY [8 votes]: There is no closed form expression, for approximations see:

Lower and upper bounds for approximation of the Kullback-Leibler divergence between Gaussian mixture models (2012)


A lower and an upper bound for the Kullback-Leibler divergence between
  two Gaussian mixtures are proposed. The mean of these bounds provides an
  approximation to the KL divergence which is shown to be equivalent to
  a previously proposed approximation in:


Approximating the Kullback Leibler Divergence Between Gaussian Mixture Models (2007)