IMPRECISE PROBABILITY: THEORIES AND APPLICATIONS

Prague, Czech Republic

16-19 July 2007

Regression is \emph{the} central concept in applied statistics for analyzing multivariate, heterogenous data: The influence of a group of variables on one other variable is quantified by the regression parameter $\beta$. In this paper, we extend standard Bayesian inference on $\beta$ in linear regression models by considering imprecise conjugated priors. Inspired by a variation and an extension of a method for inference in i.i.d.\ exponential families presented at \textsc{isipta}'05 by Quaeghebeur and de Cooman, we develop a general framework for handling linear regression models including analysis of variance models, and discuss obstacles in direct implementation of the method. Then properties of the interval-valued point estimates for a two-regressor model are derived and illustrated with simulated data. As a practical example we take a small data set from the \textsc{airgene} study and consider the influence of age and body mass index on the concentration of an inflammation marker.

** Keywords. ** \textsc{airgene} study, analysis of variance, exponential family, (imprecise) conjugate priors, imprecise probability models, interval probability, prior-data conflict, regression, robust Bayesian inference

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** Authors addresses: **

Gero Walter

c/o

Institut für Statistik,

Ludwigstr. 33

80539 München

Thomas Augustin

Department of Statistics

University of Munich

Ludwigstr. 33

D-80539 Munich

Germany

Annette Peters

GSF - Institut für Epidemiologie

Ingolstädter Landstraße 1

85764 Neuherberg

** E-mail addresses: **

Gero Walter | gero.walter@campus.lmu.de |

Thomas Augustin | thomas@stat.uni-muenchen.de |

Annette Peters | peters@gsf.de |

** Related Web Sites **

- Diploma Thesis Gero Walter

- Discussion Paper "The Normal Regression Model as a LUCK-model"

- Discussion Paper "Sketch of an Alternative Approach to Linear Regression Analysis under Sets of Conjugate Priors"

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