TITLE: What does the KL being symmetric tell us about the distributions?
QUESTION [9 upvotes]: Suppose two probability density functions, $p$ and $q$, such that $\text{KL}(q||p) = \text{KL}(p||q) \neq 0$. Intuitively, does that tell us anything interesting about the nature of these densities?

REPLY [5 votes]: One example where this happens is when $P$ and $Q$ are "antipodal" distributions -- say, $p=(p_1,\ldots,p_n)$, $q=(q_1,\ldots,q_n)$,
$q_i=1-p_i$, and 
$$P=\mathrm{Ber}(p_1)\times\mathrm{Ber}(p_2)\times\ldots
\mathrm{Ber}(p_n)$$
and $Q$ is defined analogously. Then $KL(P||Q)=KL(Q||P)$. These play a role in optimal decision theory, see
http://jmlr.org/beta/papers/v16/berend15a.html
and
https://projecteuclid.org/euclid.aos/1564797865
.

REPLY [5 votes]: We're looking at the equation $$-\sum_{x\in\mathcal{X}} P(x) \log\left(\frac{Q(x)}{P(x)}\right)
=
-\sum_{x\in\mathcal{X}} Q(x) \log\left(\frac{P(x)}{Q(x)}\right). 
$$
In the Bernoulli case,
$$-p \log\left(\frac{q}{p}\right)
-(1-p) \log\left(\frac{1-q}{1-p}\right)
=
-q \log\left(\frac{p}{q}\right)
-(1-q) \log\left(\frac{1-p}{1-q}\right)
$$
The only solutions are random variables $X$ and $Y$ with $X=1-Y$ (mentioned by @Aryeh) and the trivial solution $X=Y$: