### Abstract

The EM algorithm is a sophisticated method for estimating statistical models with hidden variables based on the Kullback-Leibler divergence. A natural extension of the Kullback-Leibler divergence is given by a class of Bregman divergences, which in general enjoy robustness to contamination data in statistical inference. In this paper, a modification of the EM algorithm based on the Bregman divergence is proposed for estimating finite mixture models. The proposed algorithm is geometrically interpreted as a sequence of projections induced from the Bregman divergence. Since a rigorous algorithm includes a nonlinear optimization procedure, two simplification methods for reducing computational difficulty are also discussed from a geometrical viewpoint. Numerical experiments on a toy problem are carried out to confirm appropriateness of the simplifications.

Original language | English |
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Pages (from-to) | 3-25 |

Number of pages | 23 |

Journal | Annals of the Institute of Statistical Mathematics |

Volume | 59 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2007 Mar |

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### Keywords

- Bregman divergence
- EM algorithm
- Finite mixture models

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability

### Cite this

**A modified em algorithm for mixture models based on Bregman divergence.** / Fujimoto, Yu; Murata, Noboru.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - A modified em algorithm for mixture models based on Bregman divergence

AU - Fujimoto, Yu

AU - Murata, Noboru

PY - 2007/3

Y1 - 2007/3

N2 - The EM algorithm is a sophisticated method for estimating statistical models with hidden variables based on the Kullback-Leibler divergence. A natural extension of the Kullback-Leibler divergence is given by a class of Bregman divergences, which in general enjoy robustness to contamination data in statistical inference. In this paper, a modification of the EM algorithm based on the Bregman divergence is proposed for estimating finite mixture models. The proposed algorithm is geometrically interpreted as a sequence of projections induced from the Bregman divergence. Since a rigorous algorithm includes a nonlinear optimization procedure, two simplification methods for reducing computational difficulty are also discussed from a geometrical viewpoint. Numerical experiments on a toy problem are carried out to confirm appropriateness of the simplifications.

AB - The EM algorithm is a sophisticated method for estimating statistical models with hidden variables based on the Kullback-Leibler divergence. A natural extension of the Kullback-Leibler divergence is given by a class of Bregman divergences, which in general enjoy robustness to contamination data in statistical inference. In this paper, a modification of the EM algorithm based on the Bregman divergence is proposed for estimating finite mixture models. The proposed algorithm is geometrically interpreted as a sequence of projections induced from the Bregman divergence. Since a rigorous algorithm includes a nonlinear optimization procedure, two simplification methods for reducing computational difficulty are also discussed from a geometrical viewpoint. Numerical experiments on a toy problem are carried out to confirm appropriateness of the simplifications.

KW - Bregman divergence

KW - EM algorithm

KW - Finite mixture models

UR - http://www.scopus.com/inward/record.url?scp=33847231874&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33847231874&partnerID=8YFLogxK

U2 - 10.1007/s10463-006-0097-x

DO - 10.1007/s10463-006-0097-x

M3 - Article

AN - SCOPUS:33847231874

VL - 59

SP - 3

EP - 25

JO - Annals of the Institute of Statistical Mathematics

JF - Annals of the Institute of Statistical Mathematics

SN - 0020-3157

IS - 1

ER -