### Abstract

We give a short proof of the following result. Let (X, Y) be any distribution on N×{0, 1}, and let (X_{1}, Y_{1}), ..., (X_{n}, Y_{n}) be an i.i.d. sample drawn from this distribution. In discrimination, the Bayes error L* = inf_{g} P{g(X)≠Y} is of crucial importance. Here we show that without further conditions on the distribution of (X, Y), no rate-of-convergence results can be obtained. Let φ_{n}(X_{1}, Y_{1}, ..., X_{n}, Y_{n}) be an estimate of the Bayes error, and let {φ_{n}(.)} be a sequence of such estimates. For any sequence {a_{n}} of positive numbers converging to zero, a distribution of (X, Y) may be found such that E{|L*-φ_{n}(X_{1}, Y_{1}, ..., X_{n}, Y_{n})|}≥a_{n} infinitely often.

Original language | English |
---|---|

Pages (from-to) | 643-645 |

Number of pages | 3 |

Journal | IEEE Transactions on Pattern Analysis and Machine Intelligence |

Volume | 21 |

Issue number | 7 |

DOIs | |

Publication status | Published - 1999 |

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### ASJC Scopus subject areas

- Control and Systems Engineering
- Electrical and Electronic Engineering
- Artificial Intelligence
- Computer Vision and Pattern Recognition

### Cite this

*IEEE Transactions on Pattern Analysis and Machine Intelligence*,

*21*(7), 643-645. https://doi.org/10.1109/34.777375

**Lower bounds for Bayes error estimation.** / Antos, András; Devroye, Luc; Györfi, L.

Research output: Contribution to journal › Article

*IEEE Transactions on Pattern Analysis and Machine Intelligence*, vol. 21, no. 7, pp. 643-645. https://doi.org/10.1109/34.777375

}

TY - JOUR

T1 - Lower bounds for Bayes error estimation

AU - Antos, András

AU - Devroye, Luc

AU - Györfi, L.

PY - 1999

Y1 - 1999

N2 - We give a short proof of the following result. Let (X, Y) be any distribution on N×{0, 1}, and let (X1, Y1), ..., (Xn, Yn) be an i.i.d. sample drawn from this distribution. In discrimination, the Bayes error L* = infg P{g(X)≠Y} is of crucial importance. Here we show that without further conditions on the distribution of (X, Y), no rate-of-convergence results can be obtained. Let φn(X1, Y1, ..., Xn, Yn) be an estimate of the Bayes error, and let {φn(.)} be a sequence of such estimates. For any sequence {an} of positive numbers converging to zero, a distribution of (X, Y) may be found such that E{|L*-φn(X1, Y1, ..., Xn, Yn)|}≥an infinitely often.

AB - We give a short proof of the following result. Let (X, Y) be any distribution on N×{0, 1}, and let (X1, Y1), ..., (Xn, Yn) be an i.i.d. sample drawn from this distribution. In discrimination, the Bayes error L* = infg P{g(X)≠Y} is of crucial importance. Here we show that without further conditions on the distribution of (X, Y), no rate-of-convergence results can be obtained. Let φn(X1, Y1, ..., Xn, Yn) be an estimate of the Bayes error, and let {φn(.)} be a sequence of such estimates. For any sequence {an} of positive numbers converging to zero, a distribution of (X, Y) may be found such that E{|L*-φn(X1, Y1, ..., Xn, Yn)|}≥an infinitely often.

UR - http://www.scopus.com/inward/record.url?scp=0032663729&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032663729&partnerID=8YFLogxK

U2 - 10.1109/34.777375

DO - 10.1109/34.777375

M3 - Article

VL - 21

SP - 643

EP - 645

JO - IEEE Transactions on Pattern Analysis and Machine Intelligence

JF - IEEE Transactions on Pattern Analysis and Machine Intelligence

SN - 0162-8828

IS - 7

ER -