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I have three RVs X, Y, and T. Is the following equation true? I(X ; Y|T) = I(Y ; X|T) Can we express the conditional mutual information as: (X;Y|T) = I(X;Y) - I(X;Y;T) ?

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    $\begingroup$ What parts of your question cannot be answered by this Wikipedia page? $\endgroup$
    – Mahmoud
    Feb 1 at 11:31

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Yes, $I(X ; Y|T)=H(X|T)+H(Y|T)-H(X,Y |T)=I(Y ; X|T)$.
Note that $H(A|B)=H(A)-I(A;B)$, so we can write $H(X|T)=H(X)-I(X;T)$ and $H(Y|T)=H(Y)-I(Y;T)$. Similarly for $H(X,Y | T)$.
Now we can collect $H(X), H(Y)$ and $H(X,Y)$ from first terms of all three to write $H(X)+H(Y)-H(X,Y)=I(X;Y)$ i.e. $$ I(X ; Y|T)=I(X;Y)-(I(X;T)+I(Y;T)-I(X,Y; T) )=I(X;Y)-I(X;Y;T) $$ Like @mhdadk pointed out, all identities are in mi and conditional mutual information

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