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Joint entropy is the information entropy of an entire random vector, treating its coordinates together as one outcome. For random variables with finite alphabets and joint probability mass function $p$,It is measured in bits. The distinction from separate information entropies is whether shared uncertainty is counted once or repeatedly. Two copies of the same fair bit have joint entropy one bit, since only $00$ and $11$ can occur. Two independent fair bits have joint entropy two bits, since all four pairs are possible and equiprobable.

For a less symmetric example, assign probabilities $1/2,1/4,1/4$ to $(X,Y)=(0,0),(1,0),(1,1)$ respectively. ThenHere $X$ is fair, while $\mathbb{P}(Y=1)=1/4$. Thus $H(X)=1$ and $H(Y)=h_2(1/4)\approx 0.811278$ bits, where $h_2$ is binary entropy. Adding the separate information entropies would give approximately $1.811278$ bits, overcounting by the mutual information $I(X;Y)\approx 0.311278$ bits.

The chain rule for information entropy makes the shared information explicit:In this example $Y$ is certain given $X=0$ and fair given $X=1$, so its average conditional entropy is $H(Y\mid X)=\frac{1}{2}\cdot 0+\frac{1}{2}\cdot 1=1/2$ bit. Sending the first coordinate changes how much uncertainty remains in the second. A prefix code with codewords 0, 10, 11 for the three possible pairs has average length $3/2$ bits, exactly the joint entropy in this stipulated law.

For finite alphabets,The lower bound follows because knowing the full random vector determines every coordinate; the upper bound follows from conditioning reduces entropy in the chain rule for information entropy. Equality in the upper bound means joint independence. The difference from that upper bound is total correlation, whose noisy parity total correlation example shows why separate pairwise comparisons need not find all the dependence.

Keep $X,Y$ as independent fair bits, introduce an independent Bernoulli random variable $N$ with $\mathbb{P}(N=1)=\epsilon$, and set $Z=X\oplus Y\oplus N$. The extra bit flips the integer parity rule with probability $\epsilon$. For each even-integer parity triple the probability is $(1-\epsilon )/4$; for each odd-integer parity triple it is $\epsilon /4$.

Every coordinate remains fair, and every pair remains independent for every $0\le \epsilon \le 1$. For example, after fixing $X$, the fair independent $Y$ makes $Y\oplus N$ fair regardless of $\epsilon$. A pair's joint probability distribution therefore does not change as noise destroys the three-way relation.

Knowing $X,Y$, uncertainty in $Z$ is exactly the uncertainty in $N$. The chain rule for information entropy giveswhere $h_2$ is binary entropy. HenceAt $\epsilon =1/4$, the result is approximately $0.188722$ bits, although all three pairwise mutual informations are zero. At $\epsilon =1/2$, all eight triples have probability $1/8$, giving joint independence and zero total correlation. At $\epsilon =1$, the complementary integer parity rule is deterministic again, so total correlation returns to one bit.

The example distinguishes "there is a dependence" from "a pair reveals it." Once $X,Y$ are known together, $Z$ is predictable at zero noise; knowing either one alone leaves a fair bit. Total correlation detects the joint departure from independence, but does not by itself specify which variables cause which others.