Wikipedia / Conditional entropy
Wikipedia / Conditional entropy
In information theory, the conditional entropy quantifies the amount of information needed to describe the outcome of a random variable is known. Here, information is measured in shannonss, natss, or hartleys.
Definition
is defined as
denote the support sets.
Note:.
Intuitively, notice that by definition of expected value and of conditional probability.
Motivation
be the entropy have probability mass function, i.e.
is the information content of the outcome is defined by:
is known in some domains as .
Given discrete random variables is defined as the weighted sum as the weights:
Properties
Conditional entropy equals zero
Conditional entropy of independent random variables
are independent random variables.
Chain rule
has joint entropy, 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.. Subtracting the two equations implies Bayes' rule.
is conditionally independent we have:
Other properties
:
is the mutual information.
:
for a given random variate.
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 be a continuous random variables with a joint probability density function 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:
are independent.
Relation to estimator error
The conditional differential entropy yields a lower bound on the expected squared error of an 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
References