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{{Short description|Measure of relative information in probability theory}}
{{Information theory}}

[[Image:Entropy-mutual-information-relative-entropy-relation-diagram.svg|thumb|256px|right|[[Venn diagram]] showing additive and subtractive relationships various [[Quantities of information|information measures]] associated with correlated variables <math>X</math> and <math>Y</math>. The area contained by both circles is the [[joint entropy]] <math>\Eta(X,Y)</math>. The circle on the left (red and violet) is the [[Entropy (information theory)|individual entropy]] <math>\Eta(X)</math>, with the red being the conditional entropy <math>\Eta(X|Y)</math>. The circle on the right (blue and violet) is <math>\Eta(Y)</math>, with the blue being <math>\Eta(Y|X)</math>. The violet is the [[mutual information]] <math>\operatorname{I}(X;Y)</math>.]]

In [[information theory]], the '''conditional entropy''' quantifies the amount of information needed to describe the outcome of a [[random variable]] <math>Y</math> given that the value of another random variable <math>X</math> is known. Here, information is measured in [[Shannon (unit)|shannon]]s, [[Nat (unit)|nat]]s, or [[Hartley (unit)|hartley]]s. The ''entropy of <math>Y</math> conditioned on <math>X</math>'' is written as <math>\Eta(Y|X)</math>.

== Definition ==
The conditional entropy of <math>Y</math> given <math>X</math> is defined as

:<math>\Eta(Y|X)\ = -\sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log \frac {p(x,y)} {p(x)}</math>

where <math>\mathcal X</math> and <math>\mathcal Y</math> denote the [[Support (mathematics)|support sets]] of <math>X</math> and <math>Y</math>.

''Note:'' Here, the convention is that the expression <math>0 \log 0</math> should be treated as being equal to zero. This is because <math>\lim_{\theta\to0^+} \theta\, \log \theta = 0</math>.<ref>{{Cite web|url=http://www.inference.org.uk/mackay/itprnn/book.html|title=David MacKay: Information Theory, Pattern Recognition and Neural Networks: The Book|website=www.inference.org.uk|access-date=2019-10-25}}</ref> 

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

== Motivation ==
Let <math>\Eta(Y|X=x)</math> be the [[Shannon Entropy|entropy]] of the discrete random variable <math>Y</math> conditioned on the discrete random variable <math>X</math> taking a certain value <math>x</math>. Denote the support sets of <math>X</math> and <math>Y</math> by <math>\mathcal X</math> and <math>\mathcal Y</math>. Let <math>Y</math> have [[probability mass function]] <math>p_Y{(y)}</math>. The unconditional entropy of <math>Y</math> is calculated as <math>\Eta(Y) := \mathbb{E}[\operatorname{I}(Y)]</math>, i.e.

:<math>\Eta(Y) = \sum_{y\in\mathcal Y} {\mathrm{Pr}(Y=y)\,\mathrm{I}(y)}
= -\sum_{y\in\mathcal Y} {p_Y(y) \log_2{p_Y(y)}},</math>

where <math>\operatorname{I}(y_i)</math> is the [[information content]] of the [[Outcome (probability)|outcome]] of <math>Y</math> taking the value <math>y_i</math>. The entropy of <math>Y</math> conditioned on <math>X</math> taking the value <math>x</math> is defined by:

:<math>\Eta(Y|X=x)
= -\sum_{y\in\mathcal Y} {\Pr(Y = y|X=x) \log_2{\Pr(Y = y|X=x)}}.</math>
Note that <math>\Eta(Y|X)</math> is the result of averaging <math>\Eta(Y|X=x)</math> over all possible values <math>x</math> that <math>X</math> may take. Also, if the above sum is taken over a sample <math>y_1, \dots, y_n</math>, the expected value <math>E_X[ \Eta(y_1, \dots, y_n \mid X = x)]</math> is known in some domains as '''{{Visible anchor|equivocation}}'''.<ref>{{cite journal|author1=Hellman, M.|author2=Raviv, J.|year=1970|title=Probability of error, equivocation, and the Chernoff bound|journal=IEEE Transactions on Information Theory|volume=16|issue=4|pages=368–372|doi=10.1109/TIT.1970.1054466|citeseerx=10.1.1.131.2865}}</ref>

Given [[Discrete random variable|discrete random variables]] <math>X</math> with image <math>\mathcal X</math> and <math>Y</math> with image <math>\mathcal Y</math>, the conditional entropy of <math>Y</math> given <math>X</math> is defined as the [[Weight function|weighted sum]] of <math>\Eta(Y|X=x)</math> for each possible value of <math>x</math>, using <math>p(x)</math> as the weights:<ref name=cover1991>{{cite book|isbn=0-471-06259-6|year=1991|authorlink1=Thomas M. Cover|author1=T. Cover|author2=J. Thomas|title=Elements of Information Theory|publisher=Wiley |url=https://archive.org/details/elementsofinform0000cove|url-access=registration}}</ref>{{rp|15}}
:<math>
\begin{align}
\Eta(Y|X)\ &\equiv \sum_{x\in\mathcal X}\,p(x)\,\Eta(Y|X=x)\\
& =-\sum_{x\in\mathcal X} p(x)\sum_{y\in\mathcal Y}\,p(y|x)\,\log_2\, p(y|x)\\
& =-\sum_{x\in\mathcal X, y\in\mathcal Y}\,p(x)p(y|x)\,\log_2\,p(y|x)\\
& =-\sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log_2 \frac {p(x,y)} {p(x)}.
\end{align}
</math>

==Properties==
===Conditional entropy equals zero===
:<math>\Eta(Y|X)=0</math> [[if and only if]] the value of <math>Y</math> is completely determined by the value of <math>X</math>.

===Conditional entropy of independent random variables===
Conversely, <math>\Eta(Y|X) = \Eta(Y)</math> if and only if <math>Y</math> and <math>X</math> are [[independent random variables]].

===Chain rule===
Assume that the combined system determined by two random variables <math>X</math> and <math>Y</math> has [[joint entropy]] <math>\Eta(X,Y)</math>, that is, we need <math>\Eta(X,Y)</math> bits of information on average to describe its exact state. Now if we first learn the value of <math>X</math>, we have gained <math>\Eta(X)</math> bits of information. Once <math>X</math> is known, we only need <math>\Eta(X,Y)-\Eta(X)</math> bits to describe the state of the whole system. This quantity is exactly <math>\Eta(Y|X)</math>, which gives the ''chain rule'' of conditional entropy:

:<math>\Eta(Y|X)\, = \, \Eta(X,Y)- \Eta(X).</math><ref name=cover1991 />{{rp|17}}

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

:<math>\begin{align}
\Eta(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]
& = \Eta(X,Y) + \sum_{x \in \mathcal X} p(x)\log (p(x) ) \\[4pt]
& = \Eta(X,Y) - \Eta(X).
\end{align}</math>

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

:<math> \Eta(X_1,X_2,\ldots,X_n) =
\sum_{i=1}^n \Eta(X_i | X_1, \ldots, X_{i-1}) </math><ref name=cover1991 />{{rp|22}}

It has a similar form to [[Chain rule (probability)|chain rule]] in [[probability theory]], except that addition instead of multiplication is used.

===Bayes' rule===
[[Bayes' rule]] for conditional entropy states
:<math>\Eta(Y|X) \,=\, \Eta(X|Y) - \Eta(X) + \Eta(Y).</math>

''Proof.'' <math>\Eta(Y|X) = \Eta(X,Y) - \Eta(X)</math> and <math>\Eta(X|Y) = \Eta(Y,X) - \Eta(Y)</math>. Symmetry entails <math>\Eta(X,Y) = \Eta(Y,X)</math>. Subtracting the two equations implies Bayes' rule.

If <math>Y</math> is [[Conditional independence|conditionally independent]] of <math>Z</math> given <math>X</math> we have:

:<math>\Eta(Y|X,Z) \,=\, \Eta(Y|X).</math>

===Other properties===
For any <math>X</math> and <math>Y</math>:
:<math>\begin{align}
\Eta(Y|X) &\le \Eta(Y) \, \\
\Eta(X,Y) &= \Eta(X|Y) + \Eta(Y|X) + \operatorname{I}(X;Y),\qquad \\
\Eta(X,Y) &= \Eta(X) + \Eta(Y) - \operatorname{I}(X;Y),\, \\
\operatorname{I}(X;Y) &\le \Eta(X),\,
\end{align}</math>

where <math>\operatorname{I}(X;Y)</math> is the [[mutual information]] between <math>X</math> and <math>Y</math>.

For independent <math>X</math> and <math>Y</math>:

:<math>\Eta(Y|X) = \Eta(Y) </math> and <math>\Eta(X|Y) = \Eta(X) \, </math>

Although the specific-conditional entropy <math>\Eta(X|Y=y)</math> can be either less or greater than <math>\Eta(X)</math> for a given [[random variate]] <math>y</math> of <math>Y</math>, <math>\Eta(X|Y)</math> can never exceed <math>\Eta(X)</math>.

== Conditional differential entropy ==
=== Definition ===
The above definition is for discrete random variables. The continuous version of discrete conditional entropy is called ''conditional differential (or continuous) entropy''. Let <math>X</math> and <math>Y</math> be a continuous random variables with a [[joint probability density function]] <math>f(x,y)</math>. The differential conditional entropy <math>h(X|Y)</math> is defined as<ref name=cover1991 />{{rp|249}}

:<math>h(X|Y) = -\int_{\mathcal X, \mathcal Y} f(x,y)\log f(x|y)\,dx dy</math>.

=== 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:
:<math>h(Y|X)\,=\,h(X,Y)-h(X)</math><ref name=cover1991 />{{rp|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:
:<math>\operatorname{I}(X,Y)=h(X)-h(X|Y)=h(Y)-h(Y|X)</math>

:<math>h(X|Y) \le h(X)</math> with equality if and only if <math>X</math> and <math>Y</math> are independent.<ref name=cover1991 />{{rp|253}}

===Relation to estimator error===
The conditional differential entropy yields a lower bound on the expected squared error of an [[estimator]]. For any Gaussian random variable <math>X</math>, observation <math>Y</math> and estimator <math>\widehat{X}</math> the following holds:<ref name=cover1991 />{{rp|255}}
:<math display="block">\mathbb{E}\left[\bigl(X - \widehat{X}{(Y)}\bigr)^2\right]
\ge \frac{1}{2\pi e}e^{2h(X|Y)}</math>

This is related to the [[uncertainty principle]] from [[quantum mechanics]].

==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.

== See also ==
* [[Entropy (information theory)]]
* [[Mutual information]]
* [[Conditional quantum entropy]]
* [[Variation of information]]
* [[Entropy power inequality]]
* [[Likelihood function]]

==References==
{{Reflist}}

[[Category:Entropy and information]]
[[Category:Information theory]]

Conditional entropy - Revision history

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