# Statistical divergence

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!

This might help as a place to start. Sgarro has generalized the Kullback-Leibler divergence $$D_{\rm KL}(Q||P)$$ to multiple distributions by introducing the average divergence
$$D’(P_1,\ldots P_k)=\frac{1}{k(k-1)}\sum_{i,j=1}^k D_{\rm KL}(P_i||P_j),$$
so what you have is related to his measure of divergence by $$D(P,Q)=3D’(P,Q,M)-\frac12D_{\rm KL}(P||Q) -\frac12D_{\rm KL}(Q||P)-\frac12D_{\rm KL}(P||M) -\frac12D_{\rm KL}(Q||M).$$ where $$M=\frac12(P+Q).$$