Mutual information measures how much observing one random variable reduces uncertainty about another.

For discrete $X$ and $Y$,

$$I(X;Y)=H(X)-H(X\mid Y).$$

By symmetry,

$$I(X;Y)=H(Y)-H(Y\mid X).$$

An equivalent expression is

$$I(X;Y)=\sum_{x,y}p(x,y)\log_2\frac{p(x,y)}{p(x)p(y)}.$$

If $X$ and $Y$ are independent, the joint distribution factors and

$$I(X;Y)=0.$$

If $Y$ determines $X$ exactly, then $H(X\mid Y)=0$ and

$$I(X;Y)=H(X).$$

In communication, $X$ can represent the transmitted symbol and $Y$ the channel observation. Mutual information then measures how much information about the input remains accessible after channel uncertainty has acted.