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[[Image:Entropy-mutual-information-relative-entropy-relation-diagram.svg|thumb|256px|right|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 (information theory)|entropy]]''' is a measure of the uncertainty associated with a set of [[random variables|variables]].
==Definition==
The joint Shannon entropy of two variables <math>X</math> and <math>Y</math> is defined as
where <math>x</math> and <math>y</math> are particular values of <math>X</math> and <math>Y</math>, respectively, <math>P(x,y)</math> is the probability of these values occurring together, and <math>P(x,y) \log_2[P(x,y)]</math> is defined to be 0 if <math>P(x,y)=0</math>.
For more than two variables <math>X_1, ..., X_n</math> this expands to
where <math>x_1,...,x_n</math> are particular values of <math>X_1,...,X_n</math>, respectively, <math>P(x_1, ..., x_n)</math> is the probability of these values occurring together, and <math>P(x_1, ..., x_n) \log_2[P(x_1, ..., x_n)]</math> is defined to be 0 if <math>P(x_1, ..., x_n)=0</math>.
==Properties==
===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 <math>X</math> and <math>Y</math> are [[statistically independent]].
The joint Shannon entropy of two variables and is defined as
where and are particular values of and , respectively, is the probability of these values occurring together, and is defined to be 0 if .
For more than two variables this expands to
where are particular values of , respectively, is the probability of these values occurring together, and is defined to be 0 if .
Properties
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 and are statistically independent.