Question

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.

Answer 1

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.

Answer 2

I believe that the lower semicontinuity of KBD is proved in Cover-Thomas Information Theory book. Also in Kullback's information theory book.