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Title: Strategies for differential shrinkage in regression with non-orthogonal designs Authors:  Christopher Hans - The Ohio State University (United States) [presenting]
Abstract: Thick-tailed mixtures of $g$ priors that mix over a single, common scale parameter have gained traction as a default prior in Bayesian regression settings. Such priors shrink all regression coefficients in the same manner and can negatively impact inference and model comparison in situations where differential shrinkage across regression coefficients is appropriate. We will review two known deficiencies of existing mixtures of $g$ priors that arise under an asymptotic regime that is motivated by the common data analytic setting where one regression coefficient is expected to be much larger than the others. The driver behind these undesirable behaviors is the use of a latent scale parameter that is common to all coefficients. Classes of block hyper-$g$ priors that employ differential shrinkage across groups of coefficients have been proposed to avoid these behaviors, however the theory underlying these priors requires that the regression design matrix has a block orthogonal structure. Extensions are described to the theory underlying the behaviors that are relevant for general, non-orthogonal designs, and introduces new prior distributions for imposing differential shrinkage in this setting. The priors rely on identifying blocks of related predictors that can be prioritized in terms of their relationship with the response. We discuss strategies for analysis that are robust to these modeling choices.