 Chain rule for mutual information

For finite discrete random variables, the mutual information supplied by two observations equals the information supplied by the first plus the additional conditional mutual information supplied by the second:The order changes the individual contributions but leaves the total unchanged.^[ref] To prove the first identity, add and subtract the intermediate conditional entropy:With base-two logarithms, all contributions are in bits. Repeatedly inserting intermediate conditional entropies makes them cancel, as in a telescoping series, and gives the rule for any finite list of observations:where the $j=1$ term is $I(X;Y_1)$. The same calculation with $W$ already known gives $I(X;Y,Z\mid W)=I(X;Y\mid W)+I(X;Z\mid Y,W)$.

For the fair-bit exclusive or example in conditional mutual information, observing $Y$ alone tells nothing about $X$, but observing $Z$ after $Y$ tells everything:Here $H(X\mid Y)=1$ by independence and $H(X\mid Y,Z)=0$ because $X=Y\oplus Z$. In the reverse order, $H(X\mid Z)=1$, so $I(X;Z)=0$ and $I(X;Y\mid Z)=1$: the two observations are individually uninformative but jointly reveal one bit. Simply adding their separate mutual informations would miss all of it.

The chain rule is useful for measuring the value of each new observation without double counting information already supplied. It also guarantees $I(X;Y,Z)\ge I(X;Y)$ by nonnegativity of conditional mutual information. Keeping $Y$ and adding $Z$ can help, even though discarding $Y$ and retaining only a processed $Z$ is constrained by the data processing inequality. Equating the two observation orders gives the precise identitywhich explains why there is no general rule that conditioning decreases mutual information.