Proposed documentation enhancement
I've seen the Dual Total Correlation formula (one of the two) be defined incorrectly, both in the method's docstring and documentation:
https://brainets.github.io/hoi/_modules/hoi/metrics/dtc.html#DTC
https://brainets.github.io/hoi/theory.html#dual-total-correlation
The formula $\sum_{j=1}^{n} H(X_j) - (n-1)H(X^{n})$ should instead use $H(X_{-j})$ (or $H(X^n_{-j})$, notation) I believe, that would fit with the original definition. Trying it on $n=3$, indeed the R.H.S. needs to equal $H(X,Y,Z)-(3\cdot H(X,Y,Z)+H(X,Y)+H(Y,Z)+H(Z,X))$ after expanding the conditions, which suggests the formula should use $H(X_{-j})$ instead of $H(X_j)$ (as that yields a $H(X)+H(Y)+H(Z)$ term instead).
I did not look into the code deeply enough to determine if it is also implemented with this mistake.
Proposed documentation enhancement
I've seen the Dual Total Correlation formula (one of the two) be defined incorrectly, both in the method's docstring and documentation:
https://brainets.github.io/hoi/_modules/hoi/metrics/dtc.html#DTC
https://brainets.github.io/hoi/theory.html#dual-total-correlation
The formula$\sum_{j=1}^{n} H(X_j) - (n-1)H(X^{n})$ should instead use $H(X_{-j})$ (or $H(X^n_{-j})$ , notation) I believe, that would fit with the original definition. Trying it on $n=3$ , indeed the R.H.S. needs to equal $H(X,Y,Z)-(3\cdot H(X,Y,Z)+H(X,Y)+H(Y,Z)+H(Z,X))$ after expanding the conditions, which suggests the formula should use $H(X_{-j})$ instead of $H(X_j)$ (as that yields a $H(X)+H(Y)+H(Z)$ term instead).
I did not look into the code deeply enough to determine if it is also implemented with this mistake.