In information theory, joint entropy is a measure of the uncertainty associated with a set of variables.[1]
X
Y
lX
lY
where
x
y
X
Y
P(x,y)
P(x,y)log2[P(x,y)]
P(x,y)=0
For more than two random variables
X1,...,Xn
where
x1,...,xn
X1,...,Xn
P(x1,...,xn)
P(x1,...,xn)log2[P(x1,...,xn)]
P(x1,...,xn)=0
The joint entropy of a set of random variables is a nonnegative number.
Η(X,Y)\geq0
Η(X1,\ldots,Xn)\geq0
The joint entropy of a set of variables is greater than or equal to the maximum of all of the individual entropies of the variables in the set.
Η(X,Y)\geqmax\left[Η(X),Η(Y)\right]
Ηl(X1,\ldots,Xnr)\geqmax1l\{Ηl(Xir)r\}
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
Y
Η(X,Y)\leqΗ(X)+Η(Y)
Η(X1,\ldots,Xn)\leqΗ(X1)+\ldots+Η(Xn)
Joint entropy is used in the definition of conditional entropy
Η(X|Y)=Η(X,Y)-Η(Y)
and
Η(X1,...,Xn)=
n | |
\sum | |
k=1 |
Η(Xk|Xk-1,...,X1)
It is also used in the definition of mutual information
\operatorname{I}(X;Y)=Η(X)+Η(Y)-Η(X,Y)
In quantum information theory, the joint entropy is generalized into the joint quantum entropy.
The above definition is for discrete random variables and just as valid in the case of continuous random variables. The continuous version of discrete joint entropy is called joint differential (or continuous) entropy. Let
X
Y
f(x,y)
h(X,Y)
For more than two continuous random variables
X1,...,Xn
The integral is taken over the support of
f
As in the discrete case the joint differential entropy of a set of random variables is smaller or equal than the sum of the entropies of the individual random variables:
h(X1,X2,\ldots,Xn)\le
n | |
\sum | |
i=1 |
h(Xi)
The following chain rule holds for two random variables:
h(X,Y)=h(X|Y)+h(Y)
h(X1,X2,\ldots,Xn)=
n | |
\sum | |
i=1 |
h(Xi|X1,X2,\ldots,Xi-1)
\operatorname{I}(X,Y)=h(X)+h(Y)-h(X,Y)