  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 also
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
mixing measures such that the resulting mixture of Pitman-Yor processes
produces 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. We explore the theoretical properties of the resulting
estimator, and show that it performs well both in simulation and in application
to real data.
