Let X be a n-dimensional Continuousrandom variable with differential entropy, h(X). The clarity of X is defined as q[X]=(1+(2πe)ne2h[X])−1.The normalizing factor (2πe)n is introduced to simplify some of the algebra. ^e46657
For any n-dimensional Continuousrandom variableX, we have for any A∈Rn×n and c∈Rn that q[X]q[X+c]q[AX]∈[0,1]=q[X]=q[X]clarity is boundedclarity is shift-invariantclarity is not scale-invariant(1)(2)(3)
\begin{proof}(1): Since h[X]∈[−∞,∞],q[X]=1+s1 for s∈[0,∞], i.e., q[X]∈[0,1]. (2),(3): Follows from properties of differential entropy i.e. Differential Entropy Under Scaling and Invariance of Differential Entropy Under Translationh[X+c]=h[X]&h[AX]=h[X]+log∣A∣\end{proof}
For any n-dimensional ContinuousrvX and any X^∈Rn, the determinant of the expected estimation error is lower-bounded as E[(X−X^)(X−X^)⊤]≥q[X]1−1with equality if and only ifX is Gaussian and E[X]=X^.
^theorem1 \begin{proof} Using the same arguments as in th 8.6.6E[(X−X^)(X−X^)⊤]≥(2πe)ne2h[X] \end{proof}