Bayesian estimation of information-theoretic metrics for sparsely sampled distributions

Abstract

Estimating the Shannon entropy of a discrete distribution from which we have only observed a small sample is challenging. Estimating other information-theoretic metrics, such as the Kullback-Leibler divergence between two sparsely sampled discrete distributions, is even harder. Existing approaches to address these problems have shortcomings: they are biased, heuristic, work only for some distributions, and/or cannot be applied to all information-theoretic metrics. Here, we propose a fast, semi-analytical estimator for sparsely sampled distributions that is efficient, precise, and general. Its derivation is grounded in probabilistic considerations and uses a hierarchical Bayesian approach to extract as much information as possible from the few observations available. Our approach provides estimates of the Shannon entropy with precision at least comparable to the state of the art, and most often better. It can also be used to obtain accurate estimates of any other information-theoretic metric, including the notoriously challenging Kullback-Leibler divergence. Here, again, our approach performs consistently better than existing estimators.

Entropy Estimators for Markovian Sequences: A Comparative Analysis

“…Finally, we stress that there are alternative entropies not considered here [ 67 ], for which the existence of accurate estimators is still an open question. Finally, an exciting possibility would be a comparative study of estimators valid for more than one random variable or probability distributions, leading, respectively, to mutual information [ 68 , 69 ] and relative entropy [ 47 , 70 , 71 ].…”

“…Even though there exists a plethora of entropy estimators in the literature [15,[39][40][41][42][43][44][45][46][47], we here focus on nine of the most commonly employed estimators, and we also propose a new estimator, constructed from known results [34].…”