# KL divergence(s) comparison,

Hi,

$P_1$, $P_2$, $P_3$ are probability distributions defined on the same support.

Knowing that $H(P_1) < H(P_2) < H(P_3)$, can we compare $D_{KL}(P_2,P_1)$ and $D_{KL}(P_3,P_1)$ ?

(H is the Shannon Entropy and $D_{KL}$ is the Kullback–Leibler divergence)

Thank you.

To be precise, if $P_1$ is not absolutely continuous w.r.t. $P_2$, then $D_{KL}(P_2,P_1)=\infty$. Similarly, $D_{KL}(P_2,P_1)=\infty$. This fact is independent of the entropies of $P_1$, $P_2$ and $P_3$. Hence, by continuity, the ratio $D_{KL}(P_2,P_1)/D_{KL}(P_3,P_1)$ can be arbitrary.
• Thank you. If we specify that KL is continuous at $(S_2, S_1)$ (respectively $(S_3, S_1)$) and that the distributions $S_1$, $S_2$, $S_3$ are strictly positive over all the support elements. Is it possible to characterize $D_{KL}(P_2,P_1)/D_{KL}(P_3,P_1)$ ? Apr 1 '13 at 15:54
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