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I have a set of random variables $(X,C) \in \mathcal{X} \times \mathcal{C}$, where $X$ can be continuous or discrete and of multiple dimensions, and $C$ is discrete.

Which are appropriate measures for comparing the similarity of the distributions $p(X|C=c)$ for $c \in \mathcal{C}$ ?

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There's a nice article in Medium article "17 types of similarity and dissimilarity measures used in data science" by Mahmoud Harmouch summarizing a large variety of statistical distance measures. Personally, I prefer staying grounded in Information Theory. The Jenson-Shannon metric begins with the Kullback-Liebler Divergence but makes it a metric by defining an intermediate distribution so that J(P,Q) = J(Q,P).

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