My goal is to find arg max of the mutual information of two variables given another variable: $$ \max_{X} \quad I(X;Y|Z) = I(X;Y) - I(X;Y;Z). $$ Is it correct to rewrite the above as: $$ \max_{X,X'} \quad I(X;Y) + \lambda I(X';Y;M)\\ \textrm{s.t.} \quad X \perp X' \;\;\; \text{or }\;\; I(X,X')=0 $$ In fact, can we optimize the $X$ with using $X'$ as auxiliary variable which is independent from $X$?
