Wrong definition (and code?) for the Dual Total Correlation

[me9hanics]

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.

[Mattehub]

Hi! Thanks for the comment. I checked the documentation and corrected the mistake. In the code, the mistake was not present.