# Conditional entropy

Individual (H(X),H(Y)), joint (H(X,Y)), and conditional entropies for a pair of correlated subsystems X,Y with mutual information I(X; Y).

In information theory, the conditional entropy (or equivocation) quantifies the remaining entropy (i.e. uncertainty) of a random variable Y given that the value of another random variable X is known. It is referred to as the entropy of Y conditional on X, and is written H(Y | X). Like other entropies, the conditional entropy is measured in bits, nats, or bans.

## Definition

More precisely, if H(Y | X = x) is the entropy of the variable Y conditional on the variable X taking a certain value x, then H(Y | X) is the result of averaging H(Y | X = x) over all possible values x that X may take.

Given discrete random variable X with support $\mathcal X$ and Y with support $\mathcal Y$, the conditional entropy of Y given X is defined as:

\begin{align} H(Y|X)\ &\equiv \sum_{x\in\mathcal X}\,p(x)\,H(Y|X=x)\\ &{=}\sum_{x\in\mathcal X}p(x)\sum_{y\in\mathcal Y}\,p(y|x)\,\log\, \frac{1}{p(y|x)}\\ &=-\sum_{x\in\mathcal X}\sum_{y\in\mathcal Y}\,p(x,y)\,\log\,p(y|x)\\ &=-\sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log\,p(y|x)\\ &=-\sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log \frac {p(x,y)} {p(x)} \\ &= \sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log \frac {p(x)} {p(x,y)}. \end{align}

Note: The supports of X and Y can be replaced by their domains if it is understood that 0log0 should be treated as being equal to zero.

## Chain rule

From this definition and the definition of conditional probability, the chain rule for conditional entropy is

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

This is true because

\begin{align} H(Y|X)=&\sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log \frac {p(x)} {p(x,y)}\\ =&-\sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log\,p(x,y) + \sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log\,p(x) \\ =& H(X,Y) + \sum_{x \in \mathcal X} p(x)\log\,p(x) \\ =& H(X,Y) - H(X). \end{align}

## Intuition

Intuitively, the combined system contains H(X,Y) bits of information: we need H(X,Y) bits of information to reconstruct its exact state. If we learn the value of X, we have gained H(X) bits of information, and the system has H(Y | X) bits of uncertainty remaining.

H(Y | X) = 0 if and only if the value of Y is completely determined by the value of X. Conversely, H(Y | X) = H(Y) if and only if Y and X are independent random variables.

## Generalization to quantum theory

In quantum information theory, the conditional entropy is generalized to the conditional quantum entropy.

## Other properties

For any X and Y:

$H(X|Y) \le H(X)$

H(X,Y) = H(X | Y) + H(Y | X) + I(X;Y), where I(X;Y) is the mutual information between X and Y.

$I(X;Y) \le H(X)$, where I(X;Y) is the mutual information between X and Y.

For independent X and Y:

H(Y | X) = H(Y) and H(X | Y) = H(X)

Although the specific-conditional entropy, H(X | Y = y), can be either lesser or greater than H(X | Y), H(X | Y = y) can never exceed H(X) when X is the uniform distribution.

## References

1. Theresa M. Korn; Korn, Granino Arthur. Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. New York: Dover Publications. pp. 613–614. ISBN 0-486-41147-8.
2. C. Arndt (2001). Information Measures: Information and its description in Science and Engineering. Berlin: Springer. pp. 370–373. ISBN 3-540-41633-1.

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