A General Scheme for Deriving Conditional Reference Priors

Reference prior algorithms are primarily available for models that have asymptotic posterior normality (i.e., regular case), barring for a specific class of models whose posteriors are not asymptotically normal (i.e., non-regular case). In particular, the current reference prior methodology is contingent on models belonging to regular or non-regular cases. This highlights that the existing refer-ence prior theory lacks a unified approach. A partial breakthrough in unifying the reference prior theory came with Berger et al. (2009) deriving an explicit form of reference prior for single group models (i.e., models with a scalar parameter or models that have all the parameters on equal footing or importance). Unfortu-nately, their approach does not generalize to multi-group models (i.e., models that have parameters' subsets ordered according to their importance). In this paper, we show that their main result can be extended with modifications to multi-group models. As a consequence, we obtain a general scheme for deriving conditional reference priors. We also prove that the invariance property of reference priors under particular transformations hold for both regular and non-regular cases. We show the usefulness of our approach by computing reference priors for models that have no known reference priors.