Joint entropy

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Individual (H(X),H(Y)), joint (H(X,Y)), and conditional entropies for a pair of correlated subsystems X,Y with mutual information I(X; Y).

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

Definition[edit]

The joint Shannon 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.

Properties[edit]

Greater than individual entropies[edit]

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)]

.

Less than or equal to the sum of individual entropies[edit]

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.

H(X,Y) \leq H(X) + H(Y)
H(X_1, ..., X_n) \leq H(X_1) + ... + H(X_n)

Relations to other entropy measures[edit]

Joint entropy is used in the definition of conditional entropy

H(X|Y) = H(Y,X) - 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.