For two signals or random variables to be independent, the mutual information (MI) must be zero.
- What does mutual information (MI) convey?
- What is the meaning of independence? Does it mean different , unrelated?
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If the mutual information is zero then the variables are independent. The closer the mutual information gets to zero the closer the variables are to being independent. If the MI is small then knowing X tells you little about X - Y.
Correlation is a linear operator. It is a weaker condition than independence. The error can be uncorrelated but nowhere close to independent.