A simple analysis of the exact probability matching prior in the location-scale model

It has long been asserted that in univariate location-scale models, when concerned with inference for either the location or scale parameter, the use of the inverse of the scale parameter as a Bayesian prior yields posterior credible sets which have exactly the correct frequentist confidence set interpretation. This claim dates to at least Peers (1965), and has subsequently been noted by various authors, with varying degrees of justification. We present a simple, direct demonstration of the exact matching property of the posterior credible sets derived under use of this prior in the univariate location-scale model. This is done by establishing an equivalence between the conditional frequentist and posterior densities of the pivotal quantities on which conditional frequentist inferences are based.