### Abstract

While deciphering the Enigma code, Good and Turing derived an unintuitive, yet effective, formula for estimating a probability distribution from a sample of data. We define the attenuation of a probability estimator as the largest possible ratio between the per-symbol probability assigned to an arbitrarily long sequence by any distribution, and the corresponding probability assigned by the estimator. We show that some common estimators have infinite attenuation and that the attenuation of the Good-Turing estimator is low, yet greater than 1. We then derive an estimator whose attenuation is 1; that is, asymptotically it does not underestimate the probability of any sequence.

Original language | English (US) |
---|---|

Pages (from-to) | 427-431 |

Number of pages | 5 |

Journal | Science |

Volume | 302 |

Issue number | 5644 |

DOIs | |

State | Published - Oct 17 2003 |

Externally published | Yes |

### ASJC Scopus subject areas

- General

### Cite this

*Science*,

*302*(5644), 427-431. https://doi.org/10.1126/science.1088284

**Always Good Turing : Asymptotically Optimal Probability Estimation.** / Orlitsky, Alon; Santhanam, Narayana P.; Zhang, Junan.

Research output: Contribution to journal › Article

*Science*, vol. 302, no. 5644, pp. 427-431. https://doi.org/10.1126/science.1088284

}

TY - JOUR

T1 - Always Good Turing

T2 - Asymptotically Optimal Probability Estimation

AU - Orlitsky, Alon

AU - Santhanam, Narayana P.

AU - Zhang, Junan

PY - 2003/10/17

Y1 - 2003/10/17

N2 - While deciphering the Enigma code, Good and Turing derived an unintuitive, yet effective, formula for estimating a probability distribution from a sample of data. We define the attenuation of a probability estimator as the largest possible ratio between the per-symbol probability assigned to an arbitrarily long sequence by any distribution, and the corresponding probability assigned by the estimator. We show that some common estimators have infinite attenuation and that the attenuation of the Good-Turing estimator is low, yet greater than 1. We then derive an estimator whose attenuation is 1; that is, asymptotically it does not underestimate the probability of any sequence.

AB - While deciphering the Enigma code, Good and Turing derived an unintuitive, yet effective, formula for estimating a probability distribution from a sample of data. We define the attenuation of a probability estimator as the largest possible ratio between the per-symbol probability assigned to an arbitrarily long sequence by any distribution, and the corresponding probability assigned by the estimator. We show that some common estimators have infinite attenuation and that the attenuation of the Good-Turing estimator is low, yet greater than 1. We then derive an estimator whose attenuation is 1; that is, asymptotically it does not underestimate the probability of any sequence.

UR - http://www.scopus.com/inward/record.url?scp=0142084741&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0142084741&partnerID=8YFLogxK

U2 - 10.1126/science.1088284

DO - 10.1126/science.1088284

M3 - Article

C2 - 14564004

AN - SCOPUS:0142084741

VL - 302

SP - 427

EP - 431

JO - Science

JF - Science

SN - 0036-8075

IS - 5644

ER -