Conditional entropy equals zero

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

Conditional entropy of independent random variables

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

Chain rule

Assume that the combined system determined by two random variables ${\displaystyle X}$ and ${\displaystyle Y}$ has joint 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 the chain 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:

${\displaystyle {\begin{aligned}\mathrm {H} (Y|X)&=\sum _{x\in {\mathcal {X}},y\in {\mathcal {Y}}}p(x,y)\log \left({\frac {p(x)}{p(x,y)}}\right)\\[4pt]&=\sum _{x\in {\mathcal {X}},y\in {\mathcal {Y}}}p(x,y)(\log(p(x))-\log(p(x,y)))\\[4pt]&=-\sum _{x\in {\mathcal {X}},y\in {\mathcal {Y}}}p(x,y)\log(p(x,y))+\sum _{x\in {\mathcal {X}},y\in {\mathcal {Y}}}{p(x,y)\log(p(x))}\\[4pt]&=\mathrm {H} (X,Y)+\sum _{x\in {\mathcal {X}}}p(x)\log(p(x))\\[4pt]&=\mathrm {H} (X,Y)-\mathrm {H} (X).\end{aligned}}}$

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

${\displaystyle \mathrm {H} (X_{1},X_{2},\ldots ,X_{n})=\sum _{i=1}^{n}\mathrm {H} (X_{i}|X_{1},\ldots ,X_{i-1})}$^[3]: 22

It has a similar form to chain rule in probability theory, except that addition instead of multiplication is used.

Bayes' rule

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 ${\displaystyle Y}$ is conditionally independent of ${\displaystyle Z}$ given ${\displaystyle X}$ we have:

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

Other properties

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

${\displaystyle {\begin{aligned}\mathrm {H} (Y|X)&\leq \mathrm {H} (Y)\,\\\mathrm {H} (X,Y)&=\mathrm {H} (X|Y)+\mathrm {H} (Y|X)+\operatorname {I} (X;Y),\qquad \\\mathrm {H} (X,Y)&=\mathrm {H} (X)+\mathrm {H} (Y)-\operatorname {I} (X;Y),\,\\\operatorname {I} (X;Y)&\leq \mathrm {H} (X),\,\end{aligned}}}$

where ${\displaystyle \operatorname {I} (X;Y)}$ is the mutual 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 given random variate ${\displaystyle y}$ of ${\displaystyle Y}$, ${\displaystyle \mathrm {H} (X|Y)}$ can never exceed ${\displaystyle \mathrm {H} (X)}$.

Definition

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

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

(Eq.2)

Properties

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 the mutual information between continuous random variables:

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

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

Relation to estimator error

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

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

This is related to the uncertainty principle from quantum mechanics.