Probability matching priors for linear calibration

In the linear calibration problem, a model is fit to paired observations arising from two measurement techniques, one known to be far more accurate (but also more expensive) than the other. The fitted model is then used with univariate observations from the less accurate technique to impute values from the more accurate one. The Bayesian paradigm emerges as attractive in this context, but the choice of an appropriate noninformative prior distribution has been controversial. In this paper we derive a class of such distributions, and provide sufficient conditions under which they lead to proper posterior densities. These priors, which we refer to as probability matching priors, are designed to produce posterior credible intervals which are asymptotically identical to their frequentist counterparts. We provide details on the implementation of our procedure using sampling-based methods, and obtain significant simplifications over previous Bayesian approaches in this area. We compare the performance of several members of our prior class in the context of two illustrative examples.