Convergence Guarantees for the Good-Turing Estimator

Amichai Painsky*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Consider a finite sample from an unknown distribution over a countable alphabet. The occupancy probability (OP) refers to the total probability of symbols that appear exactly k times in the sample. Estimating the OP is a basic problem in large alphabet modeling, with a variety of applications in machine learning, statistics and information theory. The Good-Turing (GT) framework is perhaps the most popular OP estimation scheme. Classical results show that the GT estimator converges to the OP, for every k independently. In this work we introduce new exact convergence guarantees for the GT estimator, based on worst-case mean squared error analysis. Our scheme improves upon currently known results. Further, we introduce a novel simultaneous convergence rate, for any desired set of occupancy probabilities. This allows us to quantify the unified performance of OP estimators, and introduce a novel estimation framework with favorable convergence guarantees.

Original languageEnglish
Article number279
JournalJournal of Machine Learning Research
StatePublished - 1 Sep 2022


  • Good-Turing Estimator
  • Missing Mass
  • Natural Language Modeling
  • Occupancy Probability


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