Estimating Differential Entropy under Gaussian Convolutions

Understand

This paper studies the problem of estimating the differential entropy $h(S+Z)$, where $S$ and $Z$ are independent $d$-dimensional random variables with $Z\sim\mathcal{N}(0,\sigma^2 \mathrm{I}_d)$.

• The distribution of $S$ is unknown, but $n$ independently and identically distributed (i.i.d) samples from it are available.
• The question is whether having access to samples of $S$ as opposed to samples of $S+Z$ can improve estimation performance.
• We show that the answer is positive.