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I am trying to compute the mutual information between a channel with two inputs and two outputs. I know that for a channel with 2 inputs (U,V) and 1 output (Y), the mutual information takes the following form:

\begin{equation} I(U,V;Y) = H(U,V)+H(Y) - H(U,V,Y) \end{equation}

I, on the other hand, have 2 input distributions and 2 output distributions. I also have marginal probability distributions of the inputs and the outputs and also the conditional probabilities between inputs and outputs (for convenience, I will label input distributions with $I_1$ and $I_2$ and similarly output distributions by $O_1$ and $O_2$). So What I have is the following:

\begin{equation} P(I_1),P(I_2), P(O_1), P(O_2),P(O_1,O_2|I_1,I_2). \end{equation}

What I am looking for is: \begin{equation} I(I_1,I_2;O_1,O_2)? \end{equation}

What would be the correct expression for the expression above?

Reference:

https://www3.nd.edu/~jnl/ee80653/Fall2005/tutorials/sunil.pdf https://en.wikipedia.org/wiki/Multivariate_mutual_information

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After reading William J. McGill's 1954 paper 'Multivariate Information Transmition', the answer seems to be:

\begin{equation} I(I_1,I_2;O_1,O_2) = H(I_1,I_2) + H(O_1,O_2) - H(I_1,I_2,O_1,O_2) \end{equation}

Please correct me if I am wrong.

Reference: http://link.springer.com/article/10.1007%2FBF02289159

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