The Mellin transform as a tool for prior elicitation in disease mapping

Disease mapping encompasses a set of methodologies employed to describe the disease risk distribution over a study region. When the disease under study is rare, counts are heavily affected by random variability, and the estimates of the relative risk at the small-area level are unstable. The main aim of disease mapping studies is the identification of the underlying distribution of the risk. Several approaches have been proposed for modelling unstructured and spatially structured components. After a discussion about the existing proposals for prior specification in disease mapping, we present a novel approach that allows full control on prior specification for the well-known Besag-York and Mollie' model. Our proposal is built to the aim of matching the marginal variability of the structured and unstructured components, so that priors on such components express exactly the same belief in terms of variability allocation. The goal is reached by inversion of the Mellin transform of the distribution of a quadratic form. Both an application and a simulation study are presented for comparison with other proposals