# Conditional entropy

In information theory, the **conditional entropy** (or **equivocation**) quantifies the amount of information needed to describe the outcome of a random variable given that the value of another random variable is known. Here, information is measured in shannons, nats, or hartleys. The *entropy of conditioned on * is written as .

## Definition

If is the entropy of the variable conditioned on the variable taking a certain value , then is the result of averaging over all possible values that may take.

Given discrete random variables with domain and with domain , the conditional entropy of given is defined as:^{[1]}

*Note:* It is understood that the expressions 0 log 0 and 0 log (*c*/0) for fixed *c*>0 should be treated as being equal to zero.

if and only if the value of is completely determined by the value of . Conversely, if and only if and are independent random variables.

## Chain rule

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

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

## Bayes' rule

Bayes' rule for conditional entropy states

*Proof.* and . Symmetry implies . Subtracting the two equations implies Bayes' rule. QED.

## 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.
Bayes' rule does not hold for conditional quantum entropy, since .{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
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## Other properties

where is the mutual information between and .

Although the specific-conditional entropy, , can be either less or greater than , can never exceed .

## References

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