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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 $H (X, Y)$ . Like other entropies, the joint entropy can be measured in bits, nits, or hartleys depending on the base of the logarithm.

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1 Background
2 Definition
3 Properties
4 Relations to Other Entropy Measures
5 References

Background

Given a random variable $X$ , the entropy $H (X)$ 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

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

Consider another random variable $Y$ , containing events $y$ occurring with probabilities $p y$ . $Y$ has entropy $H (Y)$ .

However, if $X$ and $Y$ describe related events, the total entropy of the system may not be $H (X) + H (Y)$ . 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 $H (X) = H (Y) = 1$ . 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 $(x, y)$ . If each pair of outcomes occurs with probability $p x, y$ , the joint entropy is defined as

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

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

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

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

Thus, the joint entropy is

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

$H(X,Y) \geq H(X)$

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

$H(X) \geq H(Y)$

Subadditivity

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.

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

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

Bounds

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

Relations to Other Entropy Measures

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

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

and the 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.

References

Theresa M. Korn; Korn, Granino Arthur. Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. New York: Dover Publications, 613-614. ISBN 0-486-41147-8.

Category: Entropy and information

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Joint_entropy". A list of authors is available in Wikipedia.