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Venn diagram for various information measures associated with correlated variables X and Y. The area contained by both circles is the joint entropy H(X,Y). The circle on the left (red and violet) is the individual entropy H(X), with the red being the conditional entropy H(X|Y). The circle on the right (blue and violet) is H(Y), with the blue being H(Y|X). The violet is the mutual information I(X;Y).

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 .

Contents

DefinitionEdit

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

Given discrete random variables   with image   and   with image  , the conditional entropy of   given   is defined as: (Intuitively, the following can be thought as the weighted sum of   for each possible value of  , using   as the weights)[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.

PropertiesEdit

Conditional entropy equals zeroEdit

  if and only if the value of   is completely determined by the value of  .

Conditional entropy of independent random variablesEdit

Conversely,   if and only if   and   are independent random variables.

Chain ruleEdit

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:

 

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

 

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

Bayes' ruleEdit

Bayes' rule for conditional entropy states

 

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

If Y is conditional independent of Z given X we have:

 

Other propertiesEdit

For any   and  :

 

where   is the mutual information between   and  .

For independent   and  :

 

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

Conditional differential entropyEdit

DefinitionEdit

The above definition is for discrete random variables and no more valid in the case of continuous 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

 .

PropertiesEdit

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 mutal information between continuous random variables:

 

Relation to estimator errorEdit

The conditional differential entropy yields a lower bound on the expected squared error of an estimator. For any random variable  , observation   and estimator   the following holds:[2]

 

Generalization to quantum theoryEdit

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  .[citation needed]

See alsoEdit

ReferencesEdit

  1. ^ Cover, Thomas M.; Thomas, Joy A. (1991). Elements of information theory (1st ed.). New York: Wiley. ISBN 0-471-06259-6. 
  2. ^ Cover, Thomas M.; Thomas, Joy A. (1991). Elements of information theory (1st ed.). New York: Wiley. ISBN 0-471-06259-6.