Conditional entropy

In information theory, the conditional entropy quantifies the amount of information needed to describe the outcome of a random variable

Here, information is measured in shannons, nats, or hartleys.

denote the support sets of

Note: Here, the convention is that the expression

[1] Intuitively, notice that by definition of expected value and of conditional probability,

with a quantity measuring the information content of

This quantity is directly related to the amount of information needed to describe the event

Hence by computing the expected value of

measures how much information, on average, the variable

be the entropy of the discrete random variable

conditioned on the discrete random variable

Denote the support sets of

have probability mass function

is the information content of the outcome of

Assume that the combined system determined by two random variables

bits of information on average to describe its exact state.

bits to describe the state of the whole system.

, 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 for conditional entropy states Proof.

Subtracting the two equations implies Bayes' rule.

The above definition is for discrete random variables.

The continuous version of discrete conditional entropy is called conditional differential (or continuous) entropy.

be a continuous random variables with a joint probability density function

is defined as[3]: 249 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:

[3]: 253 The conditional differential entropy yields a lower bound on the expected squared error of an estimator.

the following holds:[3]: 255 This is related to the uncertainty principle from quantum mechanics.

In quantum information theory, the conditional entropy is generalized to the conditional quantum entropy.

The latter can take negative values, unlike its classical counterpart.