We consider the problem of approximating the empirical Shannon entropy of a
high-frequency data stream when space limitations make exact computation
infeasible. It is known that \alpha-dependent quantities such as the Renyi and
Tsallis entropies can be estimated efficiently and unbiasedly from
low-dimensional \alpha-stable data sketches. An approximation to the Shannon
entropy can be obtained from either of these quantities by taking \alpha
sufficiently close to 1. However, practical guidelines for the choice of
$\alpha$ are lacking. We avoid this problem by going directly to the limit.