# Kullback-Leibler divergence when the $Q$ distribution has zero values

For discrete probability distributions $P,Q$, the Kullback-Leibler divergence of $Q$ from $P$ is defined to be $$D_{\mathrm{KL}} ( P \mathop{\|} Q ) = \sum_i P(i) \ln \left( \frac{P(i)}{Q(i)} \right).$$

The Kullback–Leibler divergence is defined only if $Q(i)=0$ implies $P(i)=0$, for all $i$ (absolute continuity). Whenever $P(i)$ is zero the contribution of the $i$-th term is interpreted as zero because $\lim_{x \to 0} x \ln(x) = 0$.

What if $P(i)$ is very low number and in $Q(i) = 0$, because I just don't have enough samples? If I understand it correctly, the gain of $i$-th element should be $0$?

The sentence "The Kullback–Leibler divergence is defined only if $Q(i)=0$ implies $P(i)=0$ for all $i$" implies that $D_{KL}(P\|Q)$ is not defined if there is some $i$ such that $Q(i)=0$ but $P(i)\not=0.$
One could try to finagle a definition for $D_{KL}(P\|Q)$ in these cases using limits, as is done when $P(i)=0$ for some $P(i)$. The relevant limit (using $p$ and $q$ in place of $P(i)$ and $Q(i)$) is $$\lim_{q\to 0^+}p\ln \frac{p}{q}.$$ But as $q\to0^+$, $\frac{p}{q}\to\infty$, so the limit is infinite.