How to use Kullback-leibler divergence if mean and standard deviation of of two Gaussian Distribution is provided? With Apache Spark MLLib library I am trying to find Clusters within a dataset by using Gaussian Mixture Model (number cluster =3) . Now it returns 3 different values of mean and standard deviation. I am trying to find that if there exists any overlappping  between any two distribution. To do that, I am trying find the distances between the distribution. 
Standard code for KL Div looks like this and generally takes the argument, two arrays of probabilities corresponding to two different distributions. 
Now my question is 
1. How do I change the equation to work on mean and sigma? 
2. How do I come to the conclusion that the distributions are overlapping by looking at the return value?
 A: You can compute pairwise KL divergence as a function of parameters in closed form for two Gaussian distributions $p$ and $q$. The uni-variate case:
$KL(p||q) = \log \frac{\sigma_2}{\sigma_1} + \frac{\sigma_{1}^{2} + (\mu_1-\mu_2)^2}{2\sigma_{2}^{2}} - \frac{1}{2}$
and the multi-variate case:
$KL(p||q) = \frac{1}{2}\left[\log\frac{|\Sigma_2|}{|\Sigma_1|} - d + \text{tr} (\Sigma_2^{-1}\Sigma_1) + (\mu_2 - \mu_1)^T \Sigma_2^{-1}(\mu_2 - \mu_1)\right]$
as derived here and here. Alternatively, you can try visualizing the cluster overlap by plotting the density of the mixture components.
A: To complete the answer given by Vadim, there are also many approximation of the Kullback-Leibler divergence between mixtures of Gaussian distributions.
These approximations are surprisingly easy to compute and implement. These paper by Hershey & Olsen proposes 7 or 8 different approximations and advise the use of the variational approximation: https://pdfs.semanticscholar.org/4f8d/eabc58014eae708c3e6ee27114535325067b.pdf
(Paper title is: Approximating the Kullback Leibler Divergence Between Gaussian Mixture Models.)
It will give you a similarity measure of the global mixture and you will not have to compare component by component.
