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 |
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