### Abstract

We investigate the problem of estimating the proportion vector which maximizes the likelihood of a given sample for a mixture of given densities. We adapt a framework developed for supervised learning and give simple derivations for many of the standard iterative algorithms like gradient projection and EM. In this framework, the distance between the new and old proportion vectors is used as a penalty term. The square distance leads to the gradient projection update, and the relative entropy to a new update which we call the exponentiated gradient update (EG_{η}). Curiously, when a second order Taylor expansion of the relative entropy is used, we arrive at an update EM_{η} which, for η = 1, gives the usual EM update. Experimentally, both the EM_{η}-update and the EG_{η}-update for η > 1 outperform the EM algorithm and its variants. We also prove a polynomial bound on the rate of convergence of the EG_{η} algorithm.

Original language | English (US) |
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Pages (from-to) | 97-119 |

Number of pages | 23 |

Journal | Machine Learning |

Volume | 27 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1997 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Software
- Artificial Intelligence

### Keywords

- EM
- Exponentiated gradient algorithms
- Maximum likelihood
- Mixture models

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## Cite this

*Machine Learning*,

*27*(1), 97-119. https://doi.org/10.1023/A:1007301011561