Bayesian estimation of discrete entropy with mixtures of stick-breaking priors

Evan Archer, Il Memming Park, Jonathan W. Pillow

Research output: Chapter in Book/Report/Conference proceedingConference contribution

8 Scopus citations

Abstract

We consider the problem of estimating Shannon's entropy H in the under-sampled regime, where the number of possible symbols may be unknown or countably infinite. Dirichlet and Pitman-Yor processes provide tractable prior distributions over the space of countably infinite discrete distributions, and have found major applications in Bayesian non-parametric statistics and machine learning. Here we show that they provide natural priors for Bayesian entropy estimation, due to the analytic tractability of the moments of the induced posterior distribution over entropy H. We derive formulas for the posterior mean and variance of H given data. However, we show that a fixed Dirichlet or Pitman-Yor process prior implies a narrow prior on H, meaning the prior strongly determines the estimate in the under-sampled regime. We therefore define a family of continuous mixing measures such that the resulting mixture of Dirichlet or Pitman-Yor processes produces an approximately flat prior over H. We explore the theoretical properties of the resulting estimators and show that they perform well on data sampled from both exponential and power-law tailed distributions.

Original languageEnglish (US)
Title of host publicationAdvances in Neural Information Processing Systems 25
Subtitle of host publication26th Annual Conference on Neural Information Processing Systems 2012, NIPS 2012
Pages2015-2023
Number of pages9
StatePublished - Dec 1 2012
Externally publishedYes
Event26th Annual Conference on Neural Information Processing Systems 2012, NIPS 2012 - Lake Tahoe, NV, United States
Duration: Dec 3 2012Dec 6 2012

Publication series

NameAdvances in Neural Information Processing Systems
Volume3
ISSN (Print)1049-5258

Other

Other26th Annual Conference on Neural Information Processing Systems 2012, NIPS 2012
Country/TerritoryUnited States
CityLake Tahoe, NV
Period12/3/1212/6/12

All Science Journal Classification (ASJC) codes

  • Computer Networks and Communications
  • Information Systems
  • Signal Processing

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