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Given $P(X_1, \dots, X_N)$ is there a name for the following two marginal distributions?

  • The marginal, $P(X_n)$, including only the $n$-th variable
  • The marginal, $P(X_1, \dots, X_{n-1}, X_{n+1}, \dots, X_N)$ including all but the $n$-th variable
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  • $\begingroup$ I was thinking of using "univariate marginal" for the first and "orthogonal projection" for the second marginal distribution. But maybe there is something more standard or more effective. $\endgroup$
    – Cesare
    Oct 21 '19 at 20:23
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The first I would simply call the marginal distribution of $X_n$.

I don't think the second would usually be called a marginal distribution, since the name is generally used only for univariate distributions. It's true that you've marginalized out a variable, but it's still a joint distribution. Perhaps you could call it the "joint distribution excluding $X_n$" or something like that. I don't know of a standard name. I wouldn't recommend "orthogonal projection" since (to me at least) it's unclear what that means.

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  • $\begingroup$ Actually according to this a marginal should be any subset of the $N$ variables... $\endgroup$
    – Cesare
    Oct 26 '19 at 18:29

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