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5
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What does the KL being symmetric tell us about the distributions?
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)\time …