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What are standard distance metrics for finite sets of finite sets (not necessarily power sets)? I am particularly interested in metrics which take into account the similarity between the member sets, so that sets containing similar but not identical sets are "closer" than sets containing sets which are disjoint.

For example, let $A = \{ \{1, 2, 3\}, \{0, 4, 5\} \}$, $B = \{ 1,2,3,4\}, \{ 0, 4, 6 \} \} \}$, $ C = \{ \{7, 8\}, \{4, 9\} \} $ and $D = \{ \{1,2,3\}, \{0,4,5\}, \{1, 2\} \}$. I would like $ d(A, B) \ll d(A, C)$ (because the subsets of $A$ and $B$ are more similar than the subsets of $A$ and $C$) and $d(A,B) \ll d(A,D)$ (because $|A| \neq |D|$).

Ideally, the metric $d(A,B)$ should accommodate an arbitrary distance $d'(X,Y)$ for subsets $X \subset A$ and $Y \subset B$ to measure the ``closeness'' of the subsets flexibly.

quant_dev
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  • How about $d(A,B) = \inf{|a-b|: a \in A, b \in B}$ ? – fwd Sep 16 '21 at 20:27
  • But this could be zero even if A and B were different, so not a distance metric? – quant_dev Sep 16 '21 at 20:32
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    If you do an MSE search for "metrics on power sets" you will get lots of clues. I don't know of any "standard" answer that works without some restrictions (e.g., the Hausdorff distance works only for non-empty compact sets). But I suggest you do the search. – Rob Arthan Sep 16 '21 at 20:50
  • My sets are not necessarily power sets, but they are definitely compact and can be assumed non-empty. – quant_dev Sep 17 '21 at 11:51
  • The Hausdorff distance seems useful... – quant_dev Sep 17 '21 at 12:58

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