# Joint entropy

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Joint entropy

The joint entropy is an entropy measure used in information theory. The joint entropy measures how much entropy is contained in a joint system of two random variables. If the random variables are $X$ and $Y$, the joint entropy is written $H\left(X,Y\right)$. Like other entropies, the joint entropy can be measured in bits, nits, or hartleys depending on the base of the logarithm.

Background

Given a random variable $X$, the entropy $H\left(X\right)$ describes our uncertainty about the value of $X$. If $X$ consists of several events $x$, which each occur with probability $p_x$, then the entropy of $X$ is

:$H\left(X\right) = -sum_x p_x log_2\left(p_x\right) !$

Consider another random variable $Y$, containing events $y$ occurring with probabilities $p_y$. $Y$ has entropy $H\left(Y\right)$.

However, if $X$ and $Y$ describe related events, the total entropy of the system may not be $H\left(X\right)+H\left(Y\right)$. For example, imagine we choose an integer between 1 and 8, with equal probability for each integer. Let $X$ represent whether the integer is even, and $Y$ represent whether the integer is prime. One-half of the integers between 1 and 8 are even, and one-half are prime, so $H\left(X\right)=H\left(Y\right)=1$. However, if we know that the integer is even, there is only a 1 in 4 chance that it is also prime; the distributions are related. The total entropy of the system is less than 2 bits. We need a way of measuring the total entropy of both systems.

Definition

We solve this by considering each "pair" of possible outcomes $\left(x,y\right)$. If each pair of outcomes occurs with probability $p_\left\{x,y\right\}$, the joint entropy is defined as

:$H\left(X,Y\right) = -sum_\left\{x,y\right\} p_\left\{x,y\right\} log_2\left(p_\left\{x,y\right\}\right) !$

In the example above we are not considering 1 as a prime. Then the joint probability distribution becomes:

$P\left(even,prime\right)=P\left(odd,not prime\right)=1/8 quad$

$P\left(even,not prime\right)=P\left(odd,prime\right)=3/8 quad$

Thus, the joint entropy is

$-2frac\left\{1\right\}\left\{8\right\}log_2\left(1/8\right) -2frac\left\{3\right\}\left\{8\right\}log_2\left(3/8\right) approx 1.8$ bits.

Properties

Greater than subsystem entropies

The joint entropy is always at least equal to the entropies of the original system; adding a new system can never reduce the available uncertainty.

:$H\left(X,Y\right) geq H\left(X\right)$

This inequality is an equality if and only if $Y$ is a (deterministic) function of $X$.

if $Y$ is a (deterministic) function of $X$, we also have

:$H\left(X\right) geq H\left(Y\right)$

Two systems, considered together, can never have more entropy than the sum of the entropy in each of them. This is an example of subadditivity.

:$H\left(X,Y\right) leq H\left(X\right) + H\left(Y\right)$

This inequality is an equality if and only if $X$ and $Y$ are statistically independent.

Bounds

Like other entropies, $H\left(X,Y\right) geq 0$ always.

Relations to Other Entropy Measures

The joint entropy is used in the definitions of the conditional entropy:

:$H\left(X|Y\right) = H\left(X,Y\right) - H\left(Y\right),$

and the mutual information:

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

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

References

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