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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.

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2 Answers 2

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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.

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  • $\begingroup$ Surely it is not always the case that if $(P_n,Q_n)\rightarrow (P,Q)$ then $\lim_{n\rightarrow \infty} D(P_n||Q_n)= D(P||Q)$, as you say. If $Q$ is the frontier of the PMF simplex (that is, $Q(x)=0$ for some $x$), consider a sequence $(P_n,Q_n)$ with $Q_n=Q$ and $P_n\rightarrow P$, where $P(x)=0$ whenever $Q(x)=0$. Moreover, assume all the $P_n$'s are in the interior of the simplex, that is they are all strictly positive on every $x$. Then, for each $n$, $D(P_n||Q_n)=+\infty$, hence $\lim_{n\rightarrow \infty} D(P_n||Q_n)=+\infty$, whereas $D(P||Q)<+\infty$. $\endgroup$
    – Michele
    Commented Dec 30, 2011 at 12:49
  • $\begingroup$ Yes Michele, you are right. My proof would not work in the case you have stated. But I am sure that its lower semi continuous. $\endgroup$
    – Ashok
    Commented Dec 30, 2011 at 13:27
  • $\begingroup$ For example, you can find a proof in the paper (see section III) titled "Random coding strategies for minimum entropy" published in IT Transactions. $\endgroup$
    – Ashok
    Commented Dec 30, 2011 at 13:45
  • $\begingroup$ Ashok, I found the paper, man thanks. Michele $\endgroup$
    – Michele
    Commented Dec 30, 2011 at 13:55
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I believe that the lower semicontinuity of KBD is proved in Cover-Thomas Information Theory book. Also in Kullback's information theory book.

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  • $\begingroup$ why the down-vote? $\endgroup$
    – Anthony Quas
    Commented Dec 29, 2011 at 20:59
  • $\begingroup$ Couldn't find a proof of lower semicontinuity in Cover & Thomas. $\endgroup$
    – Shlomi A
    Commented Nov 23, 2020 at 20:40

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