### Abstract

We show how information geometry throws new light on the interplay between

goodness-of-fit and estimation, a fundamental issue in statistical inference. A geometric analysis of simple, yet representative, models involving the same population parameter compellingly establishes the main theme of the paper: namely, that goodness-of-fit is necessary but not sufficient for model selection. Visual examples vividly communicate this. Specifically, for a given estimation problem, we define a class of least-informative models, linking these to both nonparametric and maximum entropy methods. Any other model is then seen to involve an informative rotation, often embodying extra-data considerations. We also look at the way that translation of models generates a form of bias-variance trade-off. Overall, our approach is a global extension of pioneering local work by Copas and Eguchi which, we note, was also geometrically inspired.

goodness-of-fit and estimation, a fundamental issue in statistical inference. A geometric analysis of simple, yet representative, models involving the same population parameter compellingly establishes the main theme of the paper: namely, that goodness-of-fit is necessary but not sufficient for model selection. Visual examples vividly communicate this. Specifically, for a given estimation problem, we define a class of least-informative models, linking these to both nonparametric and maximum entropy methods. Any other model is then seen to involve an informative rotation, often embodying extra-data considerations. We also look at the way that translation of models generates a form of bias-variance trade-off. Overall, our approach is a global extension of pioneering local work by Copas and Eguchi which, we note, was also geometrically inspired.

Original language | English |
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Title of host publication | Computational Information Geometry |

Subtitle of host publication | For Image and Signal Processing |

Editors | Frank Nielsen, Kit Dodson, Frank Critchley |

Publisher | Springer |

Pages | 63-77 |

ISBN (Electronic) | 9783319470580 |

ISBN (Print) | 9783319470566 |

Publication status | Published - 2017 |

### Publication series

Name | Signals and Communication Technology |
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### Cite this

Anaya-Izquierdo, K., Marriott, P., Vos, P., & Critchley, F. (2017). On the geometric interplay between goodness-of-fit and estimation: Illustrative examples. In F. Nielsen, K. Dodson, & F. Critchley (Eds.),

*Computational Information Geometry : For Image and Signal Processing*(pp. 63-77). (Signals and Communication Technology). Springer.