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\subsection{c. mutual \subsection{mutual  information} The concept of mutual information was introduced originally in signal analysis where it was proposed as a measure of the mutual information shared by two noisy signals.[refs] The concept is also useful in statistical physics where it is related to the entropy of the system. The theory is very well-developed and the reader is referred to almost any text on information theory for a thorough discussion of the concepts involved. We will use the definition of mutual information based on probability. Given two signals $X$ and $Y$, their mutual information is defined as  \[