For random variable X and Y, the mutual information between them is defined as:
|
I(X;Y)=∑x,yp(x,y)logp(x,y)p(x)p(y)=Elogp(X,Y)p(X)p(Y) |
which is symmetrical. Alternatively,
|
I(X;Y)=∑x,yp(x,y)logp(x|y)p(x)=Elogp(X|Y)p(X) |
Interestingly, the mutual information of a random variable with itself, called the
self-information, is the entropy of the variable, ie
I(X;X)=H(X), for
|
I(X;X)=Elogp(X,X)p(X)p(X)=Elogp(X)p(X)p(X)=−Elogp(X)=H(X) |
Proposition 2.19, provided the entropies and conditional entropies are finite:
and
Both equations can be easily verified. For example,
|
H(X)−H(X|Y)=−Elogp(X)+Elogp(X|Y)=−Elogp(X)+Elogp(X,Y)p(Y)=Elogp(X,Y)p(X)p(Y)=I(X;Y) |
Furthermore, for random variable X,Y and Z, the mutual information between X and Y conditioning on Z is defined as:
|
I(X;Y|Z)=∑x,y,zp(x,y,z)logp(x,y|z)p(x|z)p(y|z)=Elogp(X,Y|Z)p(X|Z)p(Y|Z) |
Source: Information Theory.
# posted by rot13(Unafba Pune) @ 5:55 PM
