TITLE: Why is the Kullback-Leibler divergence not symmetric?
QUESTION [17 upvotes]: As known the Kullback-Leibler Divergence:
$$\operatorname{KL}=\sum_{i=1}^n \ln(\frac{P(i)}{Q(i)})P(i)$$
is not symmetric.
I would like to know how this can be seen from the formula. I am aware that I could just try it out with exchaning Q and P for some special case, but I would like to know the mathematical reason behind it. Also, is actually here "i" the random variable?
Thanks a lot
Miau

REPLY [6 votes]: While recently studying about KL divergence, I came across the following intuitive explanation from Ian Goodfellow via an example. I hope it gives an intuitive sense as to why KL divergence is not symmetric! NIPS-2016-GAN-Tutorial