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In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference.

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What does the KL being symmetric tell us about the distributions?

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?
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