Paper abstract bibtex

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.

@ARTICLE{Archer2013a, author = {Archer, Evan and Park, Il Memming and Pillow, Jonathan}, title = {{Bayes}ian Entropy Estimation for Countable Discrete Distributions}, journal = {ArXiv e-prints}, year = {2013}, month = feb, abstract = {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.}, archiveprefix = {arXiv}, citeulike-article-id = {12071222}, citeulike-linkout-0 = {http://arxiv.org/abs/1302.0328}, citeulike-linkout-1 = {http://arxiv.org/pdf/1302.0328}, day = {2}, eprint = {1302.0328}, keywords = {bayesian, entropy-estimation, nonparametric-bayes, pitman-yor-process}, posted-at = {2013-02-25 03:18:15}, primaryclass = {cs.IT}, priority = {0}, url = {http://arxiv.org/abs/1302.0328} }

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