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The development of prior distributions for Bayesian regression hastraditionally been driven by the goal of achieving sensible model selection andparameter estimation. The formalization of properties that characterize goodperformance has led to the development and popularization of thick tailedmixtures of g priors such as the Zellner--Siow and hyper-g priors. Theproperties of a particular prior are typically illuminated under limits on thelikelihood or the prior. In this paper we introduce a new, conditionalinformation asymptotic that is motivated by the common data analysis settingwhere at least one regression coefficient is much larger than others. Weanalyze existing mixtures of g priors under this limit and reveal two newbehaviors, Essentially Least Squares (ELS) estimation and the ConditionalLindley's Paradox (CLP), and argue that these behaviors are, in general,undesirable. As the driver behind both of these behaviors is the use of asingle, latent scale parameter that is common to all coefficients, we propose ablock hyper-g prior, defined by first partitioning the covariates into groupsand then placing independent hyper-g priors on the corresponding blocks ofcoefficients. We provide conditions under which ELS and the CLP are avoided bythe new class of priors, and provide consistency results under traditionalsample size asymptotics.

Block Hyper-g Priors in Bayesian Regression