On the Geometric Interplay Between Goodness-of-Fit and Estimation: Illustrative Examples

Anaya-Izquierdo, Karim; Critchley, Frank; Marriott, Paul and Vos, Paul (2016). On the Geometric Interplay Between Goodness-of-Fit and Estimation: Illustrative Examples. In: Nielsen, Frank; Critchley, Frank and Dodson, Christopher T. J. eds. Computational Information Geometry: For Image and Signal Processing. Signals and Communication Technology. Cham: Springer, pp. 63–77.

DOI: https://doi.org/10.1007/978-3-319-47058-0_3

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.

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