# Relation between the two possible KL divergences of two distributions

Given that I know $$D\left(P\parallel Q\right)<\alpha,$$ can I say anything about $$D\left(Q\parallel P\right)$$ in terms of an upper bound on it?

Also, given this upper bound on $$D\left(P\parallel Q\right)$$, can I deduce some upper bound on $$\left|P\left(A\right)-Q\left(A\right)\right|$$ for some arbitrary event $$A$$?

The answer to your first question is no: $$D(Q||P)$$ may be however large while $$D(P||Q)$$ is however small. E.g., let $$P$$ have masses $$s$$ and $$1-s$$ at points $$0$$ and $$1$$, respectively, and let $$Q$$ have masses $$t$$ and $$1-t$$ at points $$0$$ and $$1$$, respectively, where $$0. Then $$$$D(P||Q)=s\ln\frac st+(1-s)\ln\frac{1-s}{1-t},$$$$ $$$$D(Q||P)=t\ln\frac ts+(1-t)\ln\frac{1-t}{1-s}.$$$$ Let now $$t\downarrow0$$ and $$s=te^{-1/t^2}$$. Then $$D(P||Q)\to0$$ whereas $$D(Q||P)\to\infty$$.
The answer to your second question is yes: Pinsker's inequality states that $$$$\sup_A|P(A)-Q(A)|\le\sqrt{\tfrac12\,D(P||Q)}.$$$$