The mutual information , read as “the mutual information of and ,” is defined as
where denotes the relative entropy.
Mutual information, or just information, is measured in bits if the logarithm is to the base 2, and in “nats” when using the natural logarithm.
The most obvious characteristic of mutual information is that it depends on both and . There is no information in a vacuum—information is always about something. In this case, is the information in about . As its name suggests, mutual information is symmetric, , so any information carries about , also carries about .
The definition in terms of relative entropy gives a useful interpretation of as a kind of “distance” between the joint distribution and the product distribution . Recall, however, that relative entropy is not a true distance, so this is just a conceptual tool. However, it does capture another intuitive notion of information. Remember that for independent, . Thus the relative entropy “distance” goes to zero, and we have as one would expect for independent random variables.
Recall that the entropy quantifies our uncertainty about . The last line justifies the description of entropy as “self-information.”
Mutual information, or simply information, was introduced by Shannon in his landmark 1948 paper “A Mathematical Theory of Communication.”
|Date of creation||2013-03-22 12:37:35|
|Last modified on||2013-03-22 12:37:35|
|Last modified by||drummond (72)|