I'm working with Multivariate Mutual Information (MMI), specifically with three variables $(X,Y,Z)$, applied to RNA sequences. The MMI equation that I use for three variables is based on entropy expression only:
$$MI(X,Y,Z)=H(X)+H(Y)+H(Z)-[H(X,Y)+H(Y,Z)+H(X,Z)]+H(X,Y,Z)$$
and entropy (in this equation) is defined as:
$$H(X_1, \ldots,X_n)=-\sum_{X_1} \cdots \sum_{X_n}f(x_1,\ldots,x_n)\log_2[f(x_1,\ldots,f_n)]$$
Recently, I found a paper where MMI for three variables is defined as:
$$MI(X,Y,Z)=MI(X,Y)+MI(X,Z)+MI(Y,Z)-[H(X)+H(Y)+H(Z)]+H(X,Y,Z)$$
Where MI is:
$$MI=\sum_{x,y}f(X,Y)\log_2[\frac{f(X,Y)}{f(X)f(Y)}]$$
and entropy in this equation is calculated similarly to the above one, except that there is not a negative symbol. I mean:
$$H(X_1, \ldots, X_n) = \sum_{X_1} \cdots \sum_{X_n} f(x_1, \ldots, x_n) \log_2 [f(x_1, \ldots,f_n)]$$
However, when I compute both $MI(X,Y,Z)$ with the same parameters $(X,Y,Z)$ I got different values. Therefore, looks like both $MI(X,Y,Z)$ equation are not the same. Can someone, please, check if those two equation are equal or different. May be I'm doing something wrong. Thanks for your time.