Why Is the Conditional Entropy of g(X) Given X Zero?

[PhillipKP]

Hi

I'm trying to convince myself that the conditional entropy of a function of random variable X given X is 0.

H(g(X)|X)=0

The book I'm reading (Elements of Information Theory) says that since for any particular value of X, g(X) is fixed, thus the statement is true. But I don't understand why this makes the conditional entropy 0.

Obviously I don't understand conceptually what conditional entropy really is...

Can anyone please provide some "gentle" insight into this?

 

[mXSCNT]

If we know X=x, then we can precisely infer g(X) = g(x) with probability 1, and anything else with probability 0. Hence there is no uncertainty about g(X) once we know X. Written in symbols, the previous sentence is H(g(X) | X) = 0.

 

[PhillipKP]

Ah that makes very good conceptual sense. Thank you for the short but insightful explanation.

Cheers

Phillip