Conditional entropy. In information theory, the conditional entropy 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 
The conditional entropy of  given  is defined as

where  and  denote the support sets of  and .

Note: Here, the convention is that the expression  should be treated as being equal to zero. This is because . 

Intuitively, notice that by definition of expected value and of conditional probability,  can be written as , where  is defined as . One can think of  as associating each pair  with a quantity measuring the information content of  given . This quantity is directly related to the amount of information needed to describe the event  given . Hence by computing the expected value of  over all pairs of values , the conditional entropy  measures how much information, on average, the variable  encodes about .

Motivation 
Let  be the entropy of the discrete random variable  conditioned on the discrete random variable  taking a certain value . Denote the support sets of  and  by  and . Let  have probability mass function . The unconditional entropy of  is calculated as , i.e.

where  is the information content of the outcome of  taking the value . The entropy of  conditioned on  taking the value  is defined by: 

Note that  is the result of averaging  over all possible values  that  may take. Also, if the above sum is taken over a sample , the expected value  is known in some domains as .

Given discrete random variables  with image  and  with image , the conditional entropy of  given  is defined as the weighted sum of  for each possible value of , using   as the weights:

Properties

Conditional entropy equals zero
 if and only if the value of  is completely determined by the value of .

Conditional entropy of independent random variables
Conversely,  if and only if  and  are independent random variables.

Chain rule
Assume that the combined system determined by two random variables  and  has joint entropy , that is, we need  bits of information on average 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:

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

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

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

If  is conditionally independent of  given  we have:

Other properties
For any  and :

where  is the mutual information between  and .

For independent  and :

 and 

Although the specific-conditional entropy  can be either less or greater than  for a given random variate  of ,  can never exceed .

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  and  be a continuous random variables with a joint probability density function . The differential conditional entropy  is defined as

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:

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:

 with equality if and only if  and  are independent.

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 , observation  and estimator  the following holds:

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

Entropy and information
Information theory