Alternative definition of Multivariate Mutual Information The standard mutual information (MI) is given by $$I(X;Y) = H(X) + H(Y) - H(XY)$$ which is the amount of information shared by the two random variables $X$ and $Y$.
According to wikipedia article, the multivariate mutual information (MMI) in 3D case is defined as $$I(X;Y;Z) = H(X) + H(Y) + H(Z) - H(XY) - H(XZ) - H(YZ) + H(XYZ)$$ which is the gray core of the 3-circle Venn diagram in the article.
I am interested in a related metric I derived using naive extrapolation of the 2D case. I will call it $J$ since I don't know how it is called or if it has a name.
$$J(X;Y;Z)= H(X) + H(Y) + H(Z) - H(XYZ)$$
My metric $J$ would cover the whole core, excluding only the information that is unique to each variable alone. This metric makes sense to me, because it is non-negative and is only zero if all variables are independent. 
Questions:


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*Does $J$ already have a name in the literature?

*I have some multidimensional data. My null hypothesis is that all variables are completely independent. Is $J > \epsilon(N)$ a good test for this purpose? Under null hypothesis $J = 0$, but I assume it should be compared to some correction $\epsilon(N)$ due to finite data size $N$

*I notice that $J(X;Y;Z)$ double-counts the core $I(X;Y;Z)$. Thus, perhaps it is meaningful to define the variable $K(X;Y;Z) = J(X;Y;Z)-I(X;Y;Z)$. Does this variable have a name?

 A: The functional you refer to as $J$ is called the total correlation, proposed by Watanabe (1960). It has also been called multiinformation (e.g. see Studeny and Vejnarova 1998).
Regarding independence testing, it might make sense to perform a permutation test using total correlation (TC) as the test statistic. Separately permuting the observed values of each variable preserves the empirical marginal distributions, but destroys any dependence. You can sample from the null distribution of TC (i.e. assuming independence) by repeatedly generating permuted datasets and estimating the TC for each. This can be used to calculate a p value for the TC estimated from the actual data.
But, the difficulty of estimating entropy in the multivariate case might be a concern. For some relevant citations, see this post. So, I don't know whether a permutation test based on TC would work well (this also depends on the data). At the very least, care in estimating the entropy would be needed. As a possible alternative, nonparametric tests for dependence between multiple variables have been proposed in the literature.
References:


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*Watanabe (1960). Information theoretical analysis of multivariate correlation.

*Studeny and Vejnarova (1998). The multiinformation function as a tool for measuring stochastic dependence.

