  Estimating the entropy of a discrete random variable is a fundamental problem
in information theory and related fields. This problem has many applications in
various domains, including machine learning, statistics and data compression.
Over the years, a variety of estimation schemes have been suggested. However,
despite significant progress, most methods still struggle when the sample is
small, compared to the variable's alphabet size. In this work, we introduce a
practical solution to this problem, which extends the work of McAllester and
Statos (2020). The proposed scheme uses the generalization abilities of
cross-entropy estimation in deep neural networks (DNNs) to introduce improved
entropy estimation accuracy. Furthermore, we introduce a family of estimators
for related information-theoretic measures, such as conditional entropy and
mutual information. We show that these estimators are strongly consistent and
demonstrate their performance in a variety of use-cases. First, we consider
large alphabet entropy estimation. Then, we extend the scope to mutual
information estimation. Next, we apply the proposed scheme to conditional
mutual information estimation, as we focus on independence testing tasks.
Finally, we study a transfer entropy estimation problem. The proposed
estimators demonstrate improved performance compared to existing methods in all
tested setups.
