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I’ve two discrete distributions, say $p$, $q$. I am looking for a distance measure (need not be a metric, something like KL divergence is also fine) that satisfies the following:

  1. If support of $q$ is a subset of support of $p$, then the distance should be zero.
  2. If support of $q$ contains values not in support of $p$, then the distance should indicate how far should $q$ be moved so that it’s support becomes a subset of support of $p$.
Nagabhushan S N
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    And what if $q$ cannot be moved so that its support becomes a subset? Can you just take $\sup_{x\in Q}\textrm{dist}(x,P)$, where $P,Q$ are the respective supports, which is "half" of the Hausdorff distance between $P$ and $Q$? It will give you the largest distance from points in $Q$ to $P$, and will be $0$ if $Q$ is a subset. – Conifold Feb 17 '22 at 08:38
  • Thanks! I think this should be fine. – Nagabhushan S N Feb 18 '22 at 06:29

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