### Abstract

We address the problem of estimating the probability of an observed string that is drawn i.i.d. from an unknown distribution. Motivated by models of natural language, we consider the regime in which the length of the observed string and the size of the underlying alphabet are comparably large. In this regime, the maximum likelihood distribution tends to overestimate the probability of the observed letters, so the Good-Turing probability estimator is typically used instead. We show that when used to estimate the sequence probability, the Good-Turing estimator is not consistent in this regime. We then introduce a novel sequence probability estimator that is consistent. This estimator also yields consistent estimators for other quantities of interest and a consistent universal classifier.

Original language | English (US) |
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Article number | 5773059 |

Pages (from-to) | 3207-3229 |

Number of pages | 23 |

Journal | IEEE Transactions on Information Theory |

Volume | 57 |

Issue number | 6 |

DOIs | |

State | Published - Jun 1 2011 |

### All Science Journal Classification (ASJC) codes

- Information Systems
- Computer Science Applications
- Library and Information Sciences

### Keywords

- Classification
- entropy estimation
- large alphabets
- large number of rare events (LNRE)
- natural language
- probability estimation

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## Cite this

*IEEE Transactions on Information Theory*,

*57*(6), 3207-3229. [5773059]. https://doi.org/10.1109/TIT.2011.2137210