Conditioning and differential entropy

With appropriate probability density functions and finite differential entropies, $h(X)-h(X\mid Y)=I(X;Y)\ge 0$, with equality iff independence. Differential entropy does not reverse the inequality; infinite or undefined quantities cannot justify subtraction.^[ref]

However, differential entropy can be negative and depends on the choice of coordinates. For $X$ with a uniform distribution on an interval of length $a>0$, $h(X)={\log }_{2}a$; in particular $a=1/2$ gives $-1$. Rescaling $X$ by a constant $c>0$ adds ${\log }_{2}c$ to both $h(X)$ and $h(X\mid Y)$, whenever they are finite, leaving the mutual information unchanged.^[ref]

If a continuously distributed $X$ is revealed exactly, its conditional distribution is a Dirac measure with no ordinary probability density function. The discrete identity $H(X\mid X)=0$ has no finite differential entropy counterpart. Mutual information, defined through relative entropy, is infinite here.^[ref]