Entropy power ($N(X)=(2\pi e{)}^{-1}{2}^{2h(X)/n}$)

The entropy power of an $n$-dimensional random vector with finite differential entropy in bits is the variance per coordinate of an isotropic Gaussian distribution with the same differential entropy. The normalization makes $N(X)={\sigma }^2$ when $X$ has covariance matrix ${\sigma }^2{I}_n$.

// Target: probability-and-statistics.bigb

For independent $n$-dimensional random vectors with densities and finite differential entropies, the entropy power inequality statesThe sum's differential entropy is required to be defined; a value $+\infty$ makes the inequality immediate. Equality for finite nondegenerate densities occurs for Gaussian distributions with proportional covariance matrices.

// Target: probability-and-statistics.bigb

For independent positive random variables $Y_1,Y_2$ with densities, finite differential entropies and finite ${\mu }_i=\mathbb{E}{\log }_2{Y}_i$, applying the entropy power inequality to $Z_i=\ln Y_i$ givesIndeed $h(Z_i)=h(Y_i)-{\mu }_i$, while $h(Y_1{Y}_2)=h(Z_1+Z_2)+{\mu }_1+{\mu }_2$, provided the sum's differential entropy is defined. Positivity alone does not ensure these expressions exist: $Y=e^T$ with $T$ having Cauchy distribution has undefined $\mathbb{E}{\log }_{2}Y$.

// Target: algebra.bigb