Definition of Mutual Information
Formally, the mutual information of two discrete random variables X and Y can be defined as:
where p(x,y) is the joint probability distribution function of X and Y, and and are the marginal probability distribution functions of X and Y respectively.
In the case of continuous random variables, the summation is replaced by a definite double integral:
where p(x,y) is now the joint probability density function of X and Y, and and are the marginal probability density functions of X and Y respectively.
These definitions are ambiguous because the base of the log function is not specified. To disambiguate, the function I could be parameterized as I(X,Y,b) where b is the base. Alternatively, since the most common unit of measurement of mutual information is the bit, a base of 2 could be specified.
Intuitively, mutual information measures the information that X and Y share: it measures how much knowing one of these variables reduces uncertainty about the other. For example, if X and Y are independent, then knowing X does not give any information about Y and vice versa, so their mutual information is zero. At the other extreme, if X and Y are identical then all information conveyed by X is shared with Y: knowing X determines the value of Y and vice versa. As a result, in the case of identity the mutual information is the same as the uncertainty contained in Y (or X) alone, namely the entropy of Y (or X: clearly if X and Y are identical they have equal entropy).
Mutual information is a measure of the inherent dependence expressed in the joint distribution of X and Y relative to the joint distribution of X and Y under the assumption of independence. Mutual information therefore measures dependence in the following sense: I(X; Y) = 0 if and only if X and Y are independent random variables. This is easy to see in one direction: if X and Y are independent, then p(x,y) = p(x) p(y), and therefore:
Moreover, mutual information is nonnegative (i.e. I(X;Y) ≥ 0; see below) and symmetric (i.e. I(X;Y) = I(Y;X)).
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