Does anyone know about a statistical divergence of this type?
\begin{equation}
\text{D}(P||Q) = \frac{1}{2} \left[\text{KL}(M||P) + \text{KL}(M||Q)\right]
\end{equation}
where $M = \frac{1}{2} [P+Q]$. 


This divergence is very similar to the Jensen-Shannon Divergence $\text{D}(P||Q) = \text{KL}(P||M) + \text{KL}(Q||M)$ but where the distributions in the argument of the statistical divergence appear in the second argument of the KL divergences. 


I am interested in knowing if such divergence exists in the literature and to properties of such divergence. Thanks! 



Statistical divergence
That is J-divergence, commonly just called 'symmetrised KL divergence.'


* [Kullback, Solomon, and Richard A. Leibler. "On Information and Sufficiency." *The Annals of Mathematical Statistics* 22.1 (1951): 79-86.](https://projecteuclid.org/euclid.aoms/1177729694)
* [Jeffreys, Harold. "An Invariant Form for the Prior Probability in Estimation Problems." *Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences* 186.1007 (1946): 453-461.](https://royalsocietypublishing.org/doi/abs/10.1098/rspa.1946.0056)
* [Taneja, Inder Jeet. "Information and Divergence Measures." *On Generalized Information Measures and their Applications.* Advances in Electronics and Electron Physics. Vol. 76. Academic Press, 1989. 327-413.](http://www.mtm.ufsc.br/~taneja/book/node21.html) (Also available here: <http://www.mtm.ufsc.br/~taneja/book/node21.html>)


