  The estimation of information measures of continuous distributions based on
samples is a fundamental problem in statistics and machine learning. In this
paper, we analyze estimates of differential entropy in $K$-dimensional
Euclidean space, computed from a finite number of samples, when the probability
density function belongs to a predetermined convex family $\mathcal{P}$. First,
estimating differential entropy to any accuracy is shown to be infeasible if
the differential entropy of densities in $\mathcal{P}$ is unbounded, clearly
showing the necessity of additional assumptions. Subsequently, we investigate
sufficient conditions that enable confidence bounds for the estimation of
differential entropy. In particular, we provide confidence bounds for simple
histogram based estimation of differential entropy from a fixed number of
samples, assuming that the probability density function is Lipschitz continuous
with known Lipschitz constant and known, bounded support. Our focus is on
differential entropy, but we provide examples that show that similar results
hold for mutual information and relative entropy as well.
