Mutual information

In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the "amount of information" (in units such as bits) obtained about one random variable, through the other random variable. The concept of mutual information is intricately linked to that of entropy of a random variable, a fundamental notion in information theory, that defines the "amount of information" held in a random variable.

Not limited to real-valued random variables like the correlation coefficient, MI is more general and determines how similar the joint distribution p(X,Y) is to the products of factored marginal distribution p(X)p(Y). MI is the expected value of the pointwise mutual information (PMI). The most common unit of measurement of mutual information is the bit.

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 ${\displaystyle p(x)}$ and ${\displaystyle p(y)}$ are the marginal probability distribution functions of X and Y respectively.

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