Conditional mutual information ($I(X;Y\mid Z)$)

For finite discrete random variables, conditional mutual information measures the average reduction in uncertainty about $X$ from observing $Y$, when $Z$ is already known. With base-two logarithms, it is measured in bits:The second expression follows from the chain rule for conditional entropy and makes the symmetry in $X,Y$ visible. Equivalently, compute the mutual information separately in each conditional probability distribution and then average:Here $D$ is the Kullback-Leibler divergence, also computed with base-two logarithms; zero-probability terms contribute zero. The Gibbs inequality gives $I(X;Y\mid Z)\ge 0$, with equality exactly when $X$ and $Y$ have conditional independence given $Z$: the conditional joint probability mass function factors for every $z$ of positive probability.^[ref]

For a fully computed example, take independent random variables $X,Y$ that are fair bits, and let $Z=X\oplus Y$, where $\oplus$ is exclusive or. The complete joint probability distribution has only these four positive entries:

Every pair $(x,y)$ has probability $1/4=p_X(x)p_Y(y)$, so $I(X;Y)=4(1/4){\log }_{2}1=0$. But $p_Z(0)=p_Z(1)=1/2$. Given $Z=0$, the pairs $(0,0),(1,1)$ each have conditional probability $1/2$, while both conditional marginals are fair. ThusGiven $Z=1$, the pairs $(0,1),(1,0)$ give the same calculation. ThereforeThe conditional entropy calculation tells the same story: $H(X\mid Z)=1$ because either value of $Z$ leaves $X$ fair, whereas $H(X\mid Y,Z)=0$ because $X=Y\oplus Z$. Learning the exclusive or does not reveal either bit alone; it reveals whether they agree, so learning one then determines the other.

This example distinguishes conditional independence from ordinary independence. It also disproves the claim that conditioning always reduces mutual information, despite conditioning reduces entropy. There is no general ordering between $I(X;Y\mid Z)$ and $I(X;Y)$: the example above increases it from zero to one, while taking $Y=X$ and $Z=X$ for a fair bit decreases it from one to zero. The chain rule for mutual information explains why both are possible.^[ref]

Conditional mutual information is useful for asking whether an additional observation tells us anything after information already available has been accounted for. Its zero case expresses conditional independence exactly; in a Markov chain, this is the step that turns the chain rule for mutual information into the data processing inequality.

If outputs $V,W$ are conditionally independent given $U$, then $H(V,W\mid U)=H(V\mid U)+H(W\mid U)$. Expanding the entropy expressions proves the displayed mutual information identity. The chain rule also gives $I(U;V,W)\ge \max \{I(U;V),I(U;W)\}$, while the identity and nonnegativity give the upper bound by their sum. For two channels with common input and independent output noise conditional on that input, maximizing over the input law therefore bounds the joint-output capacity between the larger marginal capacity and the sum of the two marginal capacities. Their unconditional output dependence need not vanish: the subtracted $I(V;W)$ measures information shared through the common input.

Suppose a fair input bit $X$ is independent of a publicly known channel-position label $J$. At position $j$, let the channel be a binary symmetric channel with error probability $p_j$. Forgetting $J$ gives a binary symmetric channel with error $\overline{p}=\mathbb{E}{p}_J$. ThereforeThe chain rule for mutual information and $I(X;J)=0$ show that their difference is the nonnegative conditional mutual information $I(X;J\mid Y)$. Equivalently, the inequality follows from concavity of binary entropy. For $p_0=1/5,p_1=0$ with equal position weights, the known-position value is $0.639036$ bits and the pooled value is $0.531004$ bits. These quantities answer different statistical questions.