Bayesian Entropy Estimation for Countable Discrete Distributions

Evan Archer, Il Memming Park, & Jonathan W. Pillow
Journal of Machine Learning Research 15 (Oct): 2833-2868, 2014

We consider the problem of estimating Shannon's entropy H from discrete data, in cases where the number of possible symbols is unknown or even countably infinite. The Pitman-Yor process, a generalization of Dirichlet process, provides a tractable prior distribution over the space of countably infinite discrete distributions, and has found major applications in Bayesian non-parametric statistics and machine learning. Here we show that it provides a natural family of priors for Bayesian entropy estimation, due to the fact that moments of the induced posterior distribution over H can be computed analytically. We derive formulas for the posterior mean (Bayes' least squares estimate) and variance under Dirichlet and Pitman-Yor process priors. Moreover, we show that a fixed Dirichlet or Pitman-Yor process prior implies a narrow prior distribution over H, meaning the prior strongly determines the entropy estimate in the under-sampled regime. We derive a family of continuous measures for mixing Pitman-Yor processes to produce an approximately flat prior over H. We show that the resulting "Pitman-Yor Mixture" (PYM) entropy estimator is consistent for a large class of distributions. Finally, we explore the theoretical properties of the resulting estimator, and show that it performs well both in simulation and in application to real data.

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