Shannon Entropy Estimation in ∞-Alphabets from Convergence Results: Studying Plug-In Estimators

This work addresses the problem of Shannon entropy estimation in countably infinite alphabets studying and adopting some recent convergence results of the entropy functional, which is known to be a discontinuous function in the space of probabilities in ∞-alphabets. Sufficient conditions for the convergence of the entropy are used in conjunction with some deviation inequalities (including scenarios with both finitely and infinitely supported assumptions on the target distribution). From this perspective, four plug-in histogram-based estimators are studied showing that convergence results are instrumental to derive new strong consistent estimators for the entropy. The main application of this methodology is a new data-driven partition (plug-in) estimator. This scheme uses the data to restrict the support where the distribution is estimated by finding an optimal balance between estimation and approximation errors. The proposed scheme offers a consistent (distribution-free) estimator of the entropy in ∞-alphabets and optimal rates of convergence under certain regularity conditions on the problem (finite and unknown supported assumptions and tail bounded conditions on the target distribution).