  While the importance of prior selection is well understood, establishing
guidelines for selecting priors in hierarchical models has remained an active,
and sometimes contentious, area of Bayesian methodology research. Choices of
hyperparameters for individual families of priors are often discussed in the
literature, but rarely are different families of priors compared under similar
models and hyperparameters. Using simulated data, we evaluate the performance
of inverse gamma and half-$t$ priors for estimating the standard deviation of
random effects in three hierarchical models: the 8-schools model, a random
intercepts longitudinal model, and a simple multiple outcomes model. We compare
the performance of the two prior families using a range of prior
hyperparameters, some of which have been suggested in the literature, and
others that allow for a direct comparison of pairs of half-$t$ and
inverse-gamma priors. Estimation of very small values of the random effect
standard deviation led to convergence issues especially for the half-$t$
priors. For most settings, we found that the posterior distribution of the
standard deviation had smaller bias under half-$t$ priors than under their
inverse-gamma counterparts. Inverse gamma priors generally gave similar
coverage but had smaller interval lengths than their half-$t$ prior
counterparts. Our results for these two prior families will inform prior
specification for hierarchical models, allowing practitioners to better align
their priors with their respective models and goals.
