Information theory
Entropy Differential entropy Conditional entropy Joint entropy Mutual information Directed information Conditional mutual information Relative entropy Entropy rate Limiting density of discrete points
Asymptotic equipartition property Rate–distortion theory
Shannon's source coding theorem Channel capacity Noisy-channel coding theorem Shannon–Hartley theorem
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Ininformation theory, theconditional entropy quantifies the amount of information needed to describe the outcome of arandom variable${\displaystyle Y}$ given that the value of another random variable${\displaystyle X}$ is known. Here, information is measured inshannons,nats, orhartleys. Theentropy of${\displaystyle Y}$ conditioned on${\displaystyle X}$ is written as${\displaystyle \mathrm{H}(Y|X)}$.

The conditional entropy of${\displaystyle Y}$ given${\displaystyle X}$ is defined as

${\displaystyle \mathrm{H}(Y|X)=-\sum _{x\in \mathcal{X},y\in \mathcal{Y}}p(x,y)\log \frac{p(x,y)}{p(x)}}$

where${\displaystyle \mathcal{X}}$ and${\displaystyle \mathcal{Y}}$ denote thesupport sets of${\displaystyle X}$ and${\displaystyle Y}$.

Note: Here, the convention is that the expression${\displaystyle 0\log 0}$ should be treated as being equal to zero. This is because${\displaystyle \underset{\theta \to 0^{+}}{lim}\theta \log \theta =0}$.^[1]

Intuitively, notice that by definition ofexpected value and ofconditional probability,${\displaystyle {\displaystyle H(Y|X)}}$ can be written as${\displaystyle H(Y|X)=\mathbb{E}[f(X,Y)]}$, where${\displaystyle f}$ is defined as${\displaystyle {\displaystyle f(x,y):=-\log \left(\frac{p(x,y)}{p(x)}\right)=-\log (p(y|x))}}$. One can think of${\displaystyle {\displaystyle f}}$ as associating each pair${\displaystyle {\displaystyle (x,y)}}$ with a quantity measuring the information content of${\displaystyle {\displaystyle (Y=y)}}$ given${\displaystyle {\displaystyle (X=x)}}$. This quantity is directly related to the amount of information needed to describe the event${\displaystyle {\displaystyle (Y=y)}}$ given${\displaystyle (X=x)}$. Hence by computing the expected value of${\displaystyle {\displaystyle f}}$ over all pairs of values${\displaystyle (x,y)\in \mathcal{X}\times \mathcal{Y}}$, the conditional entropy${\displaystyle {\displaystyle H(Y|X)}}$ measures how much information, on average, the variable${\displaystyle X}$ encodes about${\displaystyle Y}$.

Let${\displaystyle \mathrm{H}(Y|X=x)}$ be theentropy of the discrete random variable${\displaystyle Y}$ conditioned on the discrete random variable${\displaystyle X}$ taking a certain value${\displaystyle x}$. Denote the support sets of${\displaystyle X}$ and${\displaystyle Y}$ by${\displaystyle \mathcal{X}}$ and${\displaystyle \mathcal{Y}}$. Let${\displaystyle Y}$ haveprobability mass function${\displaystyle p_Y(y)}$. The unconditional entropy of${\displaystyle Y}$ is calculated as${\displaystyle \mathrm{H}(Y):=\mathbb{E}[\mathrm{I}(Y)]}$, i.e.

${\displaystyle \mathrm{H}(Y)=\sum _{y\in \mathcal{Y}}\mathrm{P}\mathrm{r}(Y=y)\mathrm{I}(y)=-\sum _{y\in \mathcal{Y}}{p}_Y(y){\log }_2{p}_Y(y),}$

where${\displaystyle \mathrm{I}(y_i)}$ is theinformation content of theoutcome of${\displaystyle Y}$ taking the value${\displaystyle y_i}$. The entropy of${\displaystyle Y}$ conditioned on${\displaystyle X}$ taking the value${\displaystyle x}$ is defined by:

${\displaystyle \mathrm{H}(Y|X=x)=-\sum _{y\in \mathcal{Y}}Pr(Y=y|X=x){\log }_{2}Pr(Y=y|X=x).}$

Note that${\displaystyle \mathrm{H}(Y|X)}$ is the result of averaging${\displaystyle \mathrm{H}(Y|X=x)}$ over all possible values${\displaystyle x}$ that${\displaystyle X}$ may take. Also, if the above sum is taken over a sample${\displaystyle y_1,\dots ,y_n}$, the expected value${\displaystyle E_X[\mathrm{H}(y_1,\dots ,y_n\mid X=x)]}$ is known in some domains asequivocation.^[2]

Givendiscrete random variables${\displaystyle X}$ with image${\displaystyle \mathcal{X}}$ and${\displaystyle Y}$ with image${\displaystyle \mathcal{Y}}$, the conditional entropy of${\displaystyle Y}$ given${\displaystyle X}$ is defined as theweighted sum of${\displaystyle \mathrm{H}(Y|X=x)}$ for each possible value of${\displaystyle x}$, using${\displaystyle p(x)}$ as the weights:^[3]: 15

${\displaystyle \mathrm{H}(Y|X)=0}$if and only if the value of${\displaystyle Y}$ is completely determined by the value of${\displaystyle X}$.

Conversely,${\displaystyle \mathrm{H}(Y|X)=\mathrm{H}(Y)}$ if and only if${\displaystyle Y}$ and${\displaystyle X}$ areindependent random variables.

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

${\displaystyle \mathrm{H}(Y|X)=\mathrm{H}(X,Y)-\mathrm{H}(X).}$^[3]: 17

The chain rule follows from the above definition of conditional entropy:

In general, a chain rule for multiple random variables holds:

${\displaystyle \mathrm{H}(X_1,X_2,\dots ,X_n)=\sum _{i=1}^n\mathrm{H}(X_i|X_1,\dots ,X_{i-1})}$^[3]: 22

It has a similar form tochain rule inprobability theory, except that addition instead of multiplication is used.

Bayes' rule for conditional entropy states

${\displaystyle \mathrm{H}(Y|X)=\mathrm{H}(X|Y)-\mathrm{H}(X)+\mathrm{H}(Y).}$

Proof.${\displaystyle \mathrm{H}(Y|X)=\mathrm{H}(X,Y)-\mathrm{H}(X)}$ and${\displaystyle \mathrm{H}(X|Y)=\mathrm{H}(Y,X)-\mathrm{H}(Y)}$. Symmetry entails${\displaystyle \mathrm{H}(X,Y)=\mathrm{H}(Y,X)}$. Subtracting the two equations implies Bayes' rule.

If${\displaystyle Y}$ isconditionally independent of${\displaystyle Z}$ given${\displaystyle X}$ we have:

${\displaystyle \mathrm{H}(Y|X,Z)=\mathrm{H}(Y|X).}$

For any${\displaystyle X}$ and${\displaystyle Y}$:

where${\displaystyle \mathrm{I}(X;Y)}$ is themutual information between${\displaystyle X}$ and${\displaystyle Y}$.

For independent${\displaystyle X}$ and${\displaystyle Y}$:

${\displaystyle \mathrm{H}(Y|X)=\mathrm{H}(Y)}$ and${\displaystyle \mathrm{H}(X|Y)=\mathrm{H}(X)}$

Although the specific-conditional entropy${\displaystyle \mathrm{H}(X|Y=y)}$ can be either less or greater than${\displaystyle \mathrm{H}(X)}$ for a givenrandom variate${\displaystyle y}$ of${\displaystyle Y}$,${\displaystyle \mathrm{H}(X|Y)}$ can never exceed${\displaystyle \mathrm{H}(X)}$.

The above definition is for discrete random variables. The continuous version of discrete conditional entropy is calledconditional differential (or continuous) entropy. Let${\displaystyle X}$ and${\displaystyle Y}$ be a continuous random variables with ajoint probability density function${\displaystyle f(x,y)}$. The differential conditional entropy${\displaystyle h(X|Y)}$ is defined as^[3]: 249

${\displaystyle h(X|Y)=-{\int }_{\mathcal{X},\mathcal{Y}}f(x,y)\log f(x|y)dxdy}$.

In contrast to the conditional entropy for discrete random variables, the conditional differential entropy may be negative.

As in the discrete case there is a chain rule for differential entropy:

${\displaystyle h(Y|X)=h(X,Y)-h(X)}$^[3]: 253

Notice however that this rule may not be true if the involved differential entropies do not exist or are infinite.

Joint differential entropy is also used in the definition of themutual information between continuous random variables:

${\displaystyle \mathrm{I}(X,Y)=h(X)-h(X|Y)=h(Y)-h(Y|X)}$

${\displaystyle h(X|Y)\le h(X)}$ with equality if and only if${\displaystyle X}$ and${\displaystyle Y}$ are independent.^[3]: 253

The conditional differential entropy yields a lower bound on the expected squared error of anestimator. For any Gaussian random variable${\displaystyle X}$, observation${\displaystyle Y}$ and estimator${\displaystyle \hat{X}}$ the following holds:^[3]: 255

$${\displaystyle \mathbb{E}\left[(X-\hat{X}(Y){)}^2\right]\ge \frac{1}{2\pi e}{e}^{2h(X|Y)}}$$

This is related to theuncertainty principle fromquantum mechanics.

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

• Entropy (information theory)
• Mutual information
• Conditional quantum entropy
• Variation of information
• Entropy power inequality
• Likelihood function

• Entropy and information
• Information theory

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