Joint entropy

Definition

The joint Shannon entropy (in bits) of two variables X and Y is defined as

where x and y are particular values of X and Y , respectively, P ( x , y ) is the joint 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

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 .

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.

Less than or equal to the 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 of subadditivity. This inequality is an equality if and only if X and Y are statistically independent.

Relations to other entropy measures

Joint entropy is used in the definition of conditional entropy

and

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