Joint entropy

Venn diagram for various information measures associated with correlated variables X and Y. The area contained by both circles is the joint entropy H(X,Y). The circle on the left (red and violet) is the individual entropy H(X), with the red being the conditional entropy H(X|Y). The circle on the right (blue and violet) is H(Y), with the blue being H(Y|X). The violet is the mutual information I(X;Y).

In information theory, joint entropy is a measure of the uncertainty associated with a set of variables.

Definition

The joint Shannon entropy (in bits) of two variables $X$ and $Y$ is defined as

$H(X,Y) = -\sum_{x} \sum_{y} P(x,y) \log_2[P(x,y)] \!$

where $x$ and $y$ are particular values of $X$ and $Y$, respectively, $P(x,y)$ is the joint probability of these values occurring together, and $P(x,y) \log_2[P(x,y)]$ is defined to be 0 if $P(x,y)=0$.

For more than two variables $X_1, ..., X_n$ this expands to

$H(X_1, ..., X_n) = -\sum_{x_1} ... \sum_{x_n} P(x_1, ..., x_n) \log_2[P(x_1, ..., x_n)] \!$

where $x_1,...,x_n$ are particular values of $X_1,...,X_n$, respectively, $P(x_1, ..., x_n)$ is the probability of these values occurring together, and $P(x_1, ..., x_n) \log_2[P(x_1, ..., x_n)]$ is defined to be 0 if $P(x_1, ..., x_n)=0$.

Properties

Greater than individual entropies

The joint entropy of a set of variables is greater than or equal to all of the individual entropies of the variables in the set.

$H(X,Y) \geq \max[H(X),H(Y)]$

$H(X_1, ..., X_n) \geq \max[H(X_1), ..., H(X_n)]$

Less than or equal to the sum of individual entropies

The joint entropy of a set of variables is less than or equal to the sum of the individual entropies of the variables in the set. This is an example of subadditivity. This inequality is an equality if and only if $X$ and $Y$ are statistically independent.

$H(X,Y) \leq H(X) + H(Y)$

$H(X_1, ..., X_n) \leq H(X_1) + ... + H(X_n)$

Relations to other entropy measures

Joint entropy is used in the definition of conditional entropy

$H(X|Y) = H(Y,X) - H(Y)\,$,

and ${\displaystyle H(X_1,\dots,X_n) = \sum_{k=1}^N H(X_k|X_{k-1},\dots, X_1)}$It is also used in the definition of mutual information

$I(X;Y) = H(X) + H(Y) - H(X,Y)\,$

In quantum information theory, the joint entropy is generalized into the joint quantum entropy.