  The development of prior distributions for Bayesian regression has
traditionally been driven by the goal of achieving sensible model selection and
parameter estimation. The formalization of properties that characterize good
performance has led to the development and popularization of thick tailed
mixtures of g priors such as the Zellner--Siow and hyper-g priors. The
properties of a particular prior are typically illuminated under limits on the
likelihood or the prior. In this paper we introduce a new, conditional
information asymptotic that is motivated by the common data analysis setting
where at least one regression coefficient is much larger than others. We
analyze existing mixtures of g priors under this limit and reveal two new
behaviors, Essentially Least Squares (ELS) estimation and the Conditional
Lindley's Paradox (CLP), and argue that these behaviors are, in general,
undesirable. As the driver behind both of these behaviors is the use of a
single, latent scale parameter that is common to all coefficients, we propose a
block hyper-g prior, defined by first partitioning the covariates into groups
and then placing independent hyper-g priors on the corresponding blocks of
coefficients. We provide conditions under which ELS and the CLP are avoided by
the new class of priors, and provide consistency results under traditional
sample size asymptotics.
