Joint entropy

The joint Shannon entropy (in bits) of two discrete random variables ${\displaystyle X}$ and ${\displaystyle Y}$ with images ${\displaystyle {\mathcal {X}}}$ and ${\displaystyle {\mathcal {Y}}}$ is defined as^[2]: 16

${\displaystyle \mathrm {H} (X,Y)=-\sum _{x\in {\mathcal {X}}}\sum _{y\in {\mathcal {Y}}}P(x,y)\log _{2}[P(x,y)]}$

where ${\displaystyle x}$ and ${\displaystyle y}$ are particular values of ${\displaystyle X}$ and ${\displaystyle Y}$, respectively, ${\displaystyle P(x,y)}$ is the joint probability of these values occurring together, and ${\displaystyle P(x,y)\log _{2}[P(x,y)]}$ is defined to be 0 if ${\displaystyle P(x,y)=0}$.

For more than two random variables ${\displaystyle X_{1},...,X_{n}}$ this expands to

${\displaystyle \mathrm {H} (X_{1},...,X_{n})=-\sum _{x_{1}\in {\mathcal {X}}_{1}}...\sum _{x_{n}\in {\mathcal {X}}_{n}}P(x_{1},...,x_{n})\log _{2}[P(x_{1},...,x_{n})]}$

where ${\displaystyle x_{1},...,x_{n}}$ are particular values of ${\displaystyle X_{1},...,X_{n}}$, respectively, ${\displaystyle P(x_{1},...,x_{n})}$ is the probability of these values occurring together, and ${\displaystyle P(x_{1},...,x_{n})\log _{2}[P(x_{1},...,x_{n})]}$ is defined to be 0 if ${\displaystyle P(x_{1},...,x_{n})=0}$.

The joint entropy of a set of random variables is a nonnegative number.

${\displaystyle \mathrm {H} (X,Y)\geq 0}$

${\displaystyle \mathrm {H} (X_{1},\ldots ,X_{n})\geq 0}$

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

${\displaystyle \mathrm {H} (X,Y)\geq \max \left[\mathrm {H} (X),\mathrm {H} (Y)\right]}$

${\displaystyle \mathrm {H} {\bigl (}X_{1},\ldots ,X_{n}{\bigr )}\geq \max _{1\leq i\leq n}{\Bigl \{}\mathrm {H} {\bigl (}X_{i}{\bigr )}{\Bigr \}}}$

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 ${\displaystyle X}$ and ${\displaystyle Y}$ are statistically independent.^[2]: 30

${\displaystyle \mathrm {H} (X,Y)\leq \mathrm {H} (X)+\mathrm {H} (Y)}$

${\displaystyle \mathrm {H} (X_{1},\ldots ,X_{n})\leq \mathrm {H} (X_{1})+\ldots +\mathrm {H} (X_{n})}$