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On the post A formal definition of a “measure of association” @kjetil b halvorsen commented the following:

A copula could be seen as a characterization of association, so maybe a "measure of association" is a functional of the bivariate distribution (,) that only depends on the copula (,).

From Wikipedia:

[A map] $C: [0,1]^d \mapsto [0,1]$ is a $d$-dimensional copula if $C$ is a joint distribution of ad $d$-dimensional random vector on the unit cube $[0,1]^d$ with uniform marginals.

With Sklar's theorem I can see how this is a useful mathematical tool for studying distributions in general, but it hasn't hit me yet why the copula is a characterization of association. At this moment it seems more like a multivariate CDF with normalized support, and it isn't clear to me that a functional of a CDF which depends on the copula would necessarily be a measure of association.

Just as mutual independence for a collection of random variables $\{X_j\}_{j=1}^{n}$ can be given by the satisfying of the equality $F_{X_1, \cdots, X_n}(x_1,\cdots,x_n) = \prod_{j}^{n} F_{x_j}(x_j)$, do we simply rephrase this in terms of the copulas?

$$C_{X_1, \cdots, X_n}(x_1,\cdots,x_n) = \prod_{j}^{n} C_{x_j}(x_j)$$

How can a copula be seen as a characterization of association?

Footnote: There are some related questions that are interesting but do not address my question.

DifferentialPleiometry
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    I posted two questions last year that are possibly of interest: [1](https://stats.stackexchange.com/questions/510992/copula-entropy-calculation-is-borked) [2](https://stats.stackexchange.com/questions/511088/mutual-information-relationship-to-copula-entropy-is-borked). – Dave Jul 19 '21 at 13:18
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    Oops, I posted those earlier this year. Anyway, the key point is that, if we can rectify what I question in those two posts (what I call "borked" about the relationship between mutual information, copula, and copula entropy calculation), then the copula has an intimate relationship with the mutual information. – Dave Jul 19 '21 at 13:35
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    I hate to say it, but isn't this a trivial observation? The copula factors out the univariate marginal distributions, which *obviously* have nothing to do with any concept of "association." Thus, any measure of association perforce is some functional of the copula. – whuber Jul 19 '21 at 13:39
  • @whuber It may seem [trivial](https://en.wikipedia.org/wiki/Triviality_(mathematics)) [*prima facie*](https://en.wikipedia.org/wiki/Prima_facie), but I've asked in case I missed something counter-intuitive to me. – DifferentialPleiometry Jul 19 '21 at 13:45
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    @whuber: It is not that trivial, the Pearson correlation is not a functional of the copula ... – kjetil b halvorsen Jul 19 '21 at 22:52
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    @Kjetil That's an interesting observation. Maybe it says something important about Pearson correlation! – whuber Jul 20 '21 at 12:49
  • If Pearson's R is a measure of association, and Pearson's R is not a functional of the copula, then not all measures of association are a functional of a copula. – DifferentialPleiometry Jul 20 '21 at 13:47
  • Copula can measure the dependency through its dependency parameters. – Alice Jul 22 '21 at 13:22

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