Bayesian Inference for Questionable Data

In this paper we develop Bayesian procedures for vague data. The data are assumed to be vague in the sense that the likelihood is a mixture of the model distribution and an error distribution. In this case the standard updating procedure of the model prior would fail. As a new method to deal with such imprecise data we consider observable uncertainties. In this model a specified degree of belief for the validity of the observation is added to the original measurement. Our proposal involves the idea that occasionally the observations are caused by an unknown error distribution. We discuss the effect of this assumption and show a parametrical and non-parametrical analysis in this setup. For the analysis of the error distribution we establish a nonparametrical approach. Convex optimization procedures can be applied for a nonparametric estimation of the error distribution. An equivalence theorem characterizes optimal estimates and provides an iterative procedure converging to the empirical Bayes estimate.