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? 



KL divergence and mixture of Gaussians
There is no closed form expression, for approximations see:


* [Lower and upper bounds for approximation of the Kullback-Leibler divergence between Gaussian mixture models](https://infoscience.epfl.ch/record/174055/files/durrieuThiranKelly_kldiv_icassp2012_R1.pdf) (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](https://www.researchgate.net/publication/4249249_Approximating_the_Kullback_Leibler_Divergence_Between_Gaussian_Mixture_Models) (2007)


