Conditional entropy

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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 amount of information needed to describe the outcome of a random variable $Y$ given that the value of another random variable $X$ is known. Here, information is measured in bits, nats, or bans. The entropy of $Y$ conditioned on $X$ is written as $H(Y|X)$ .

Definition

If $H(Y|X=x)$ is the entropy of the variable $Y$ conditioned 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:^[1]

${\begin{aligned}H(Y|X)\ &\equiv \sum _{{x\in {\mathcal X}}}\,p(x)\,H(Y|X=x)\\&{=}\sum _{{x\in {\mathcal X}}}\left(p(x)\sum _{{y\in {\mathcal Y}}}\,p(y|x)\,\log \,{\frac {1}{p(y|x)}}\right)\\&=-\sum _{{x\in {\mathcal X}}}\sum _{{y\in {\mathcal Y}}}\,p(x,y)\,\log \,p(y|x)\\&=-\sum _{{x\in {\mathcal X},y\in {\mathcal Y}}}p(x,y)\log \,p(y|x)\\&=\sum _{{x\in {\mathcal X},y\in {\mathcal Y}}}p(x,y)\log {\frac {p(x)}{p(x,y)}}.\\\end{aligned}}$

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

$H(Y|X)=0$ 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.

Chain rule

Assume that the combined system determined by two random variables X and Y has entropy $H(X,Y)$ , that is, we need $H(X,Y)$ bits of information to describe its exact state. Now if we first learn the value of $X$ , we have gained $H(X)$ bits of information. Once $X$ is known, we only need $H(X,Y)-H(X)$ bits to describe the state of the whole system. This quantity is exactly $H(Y|X)$ , which gives the chain rule of conditional entropy:

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

Formally, the chain rule indeed follows from the above definition of conditional entropy:

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

Generalization to quantum theory

In quantum information theory, the conditional entropy is generalized to the conditional quantum entropy. The latter can take negative values, unlike its classical counterpart.

Other properties

For any $X$ and $Y$ :

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

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

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

$I(X;Y)\leq H(X),\,$

where $I(X;Y)$ is the mutual information between $X$ and $Y$ .

For independent $X$ and $Y$ :

$H(Y|X)=H(Y){\text{ and }}H(X|Y)=H(X)\,$

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

References

↑ Thomas, Thomas M. Cover, Joy A. (1991). Elements of information theory (99th ed. ed.). New York: Wiley. ISBN 0-471-06259-6.