Prior distributions for variance parameters in hierarchical models

TLDR: In this paper, a folded-noncentral-t family of conditionally conjugate priors for hierarchical standard deviation pa- rameters is proposed, and weakly informative priors in this family are considered.

Abstract: Various noninformative prior distributions have been suggested for scale parameters in hierarchical models. We construct a new folded-noncentral-t family of conditionally conjugate priors for hierarchical standard deviation pa- rameters, and then consider noninformative and weakly informative priors in this family. We use an example to illustrate serious problems with the inverse-gamma family of "noninformative" prior distributions. We suggest instead to use a uni- form prior on the hierarchical standard deviation, using the half-t family when the number of groups is small and in other settings where a weakly informative prior is desired. We also illustrate the use of the half-t family for hierarchical modeling of multiple variance parameters such as arise in the analysis of variance.