Bayesian Entropy Estimation for Countable Discrete Distributions

Evan Archer; Il Memming Park; Jonathan W. Pillow

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.

Bayesian Entropy Estimation for Countable Discrete Distributions

Abstract