Dan Xie: Joint entropy

http://en.wikipedia.org/wiki/Joint_entropy

Joint entropy is a measure of the uncertainty associated with a set of variables.

[edit]Definition

The joint entropy of two variables

X

and

Y

is defined as

$H(X,Y) = -\sum_{x} \sum_{y} P(x,y) \log_2[P(x,y)] \!$

where

x

and

y

are particular values of

X

and

Y

, respectively,

P (x, y)

is the probability of these values occurring together, and

P (x, y)log 2 [P (x, y)]

is defined to be 0 if

P (x, y) = 0

For more than two variables

X 1,..., X n

this expands to

$H(X_1, ..., X_n) = -\sum_{x_1} ... \sum_{x_n} P(x_1, ..., x_n) \log_2[P(x_1, ..., x_n)] \!$

where

x 1,..., x n

are particular values of

X 1,..., X n

, respectively,

P (x 1,..., x n)

is the probability of these values occurring together, and

P (x 1,..., x n)log 2 [P (x 1,..., x n)]

is defined to be 0 if

P (x 1,..., x n) = 0

[edit]Properties

[edit]Greater than individual entropies

The joint entropy of a set of variables is greater than or equal to all of the individual entropies of the variables in the set.

$H(X,Y) \geq max[H(X),H(Y)]$

$H(X_1, ..., X_n) \geq max[H(X_1), ..., H(X_n)]$

[edit]Less than sum of individual entropies

The joint entropy of a set of variables is less than or equal to the sum of the individual entropies of the variables in the set. This is an example ofsubadditivity. This inequality is an equality if and only if

X

and

Y

are statistically independent.

$H(X,Y) \leq H(X) + H(Y)$

$H(X_1, ..., X_n) \leq H(X_1) + ... + H(X_n)$

[edit]Relations to Other Entropy Measures

Joint entropy is used in the definition of conditional entropy --

$H(X|Y) = H(X,Y) - H(Y)\,$

-- and mutual information:

$I(X;Y) = H(X) + H(Y) - H(X,Y)\,$

In quantum information theory, the joint entropy is generalized into the joint quantum entropy.

Dan Xie

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Sunday, March 27, 2011

Joint entropy

Contents

[edit]Definition

[edit]Properties

[edit]Greater than individual entropies

[edit]Less than sum of individual entropies

[edit]Relations to Other Entropy Measures

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