Lower semicontinuity of Kullback-Leibler divergence The Kullback-Leibler Divergence (KLD) of two PMF's $P(x)$ and $Q(x)$ is $D(P||Q)=\sum_x P(x)\log(P(x)/Q(x))$, with the provisos that $0\cdot \log (0/p)=0$ and $p\cdot \log (p/0)=+\infty$ whenever $p>0$.
It is known that KLD is continuous at $(P,Q)$ if $Q$ is strictly positive over all $x$'s.  What can be said otherwise?
To be more specific, assume we are given a sequence of PMF $\{(P_n,Q_n)\}_{n\geq 0}$ s.t. $(P_n,Q_n)\rightarrow (P,Q)$ in the simplex of PFM's (with the topology induced by, say, norm-1 distance).
Is it correct to deduce that
$\lim \inf_{n\rightarrow \infty} D(P_n||Q_n) \geq D(P||Q)$ ?
This would follows if KLD is lower-semicontinuous, right?
Many thanks.
 A: In addition to the conventions you have mentioned, it is also assumed that $0\log(0/0)=0$.
With these conventions, I think, in the finite case, it is always true that $$\lim_{n\to \infty} D(P_n||Q_n)=D(P||Q)$$ As you said, if $Q(x)>0$ for all $x$, its immediate from the Dominated Convergence theorem.
The problem is only when for some $y$, $Q(y)=0$ whereas $P(y)>0$.
In which case $P(y)\log(P(y)/Q(y))=\infty$ and $D(P\|Q)=\infty$
But since $(P_n,Q_n)\to (P,Q)$, we have $P_n(y)\to P(y)$ and $Q_n(y)\to Q(y)$, whence $$P_n(y)\log(P_n(y)/Q_n(y))\to P(y)\log(P(y)/Q(y))=\infty.$$ Hence $D(P_n||Q_n)\to \infty$
So in any case we have $\lim_{n\to \infty} D(P_n||Q_n)=D(P||Q)$.
In a general measurable space (i.e., if $P_n, Q_n, P, Q$ are probability measures on some general measure space say $(\mathbb{X}, \mathcal{X})$), I think, we have only lower semicontinuity.
Pardon me, if something is wrong.
A: I believe that the lower semicontinuity of KBD is proved in Cover-Thomas Information Theory book. Also in Kullback's information theory book.
