135 research outputs found
On the half-Cauchy prior for a global scale parameter
This paper argues that the half-Cauchy distribution should replace the
inverse-Gamma distribution as a default prior for a top-level scale parameter
in Bayesian hierarchical models, at least for cases where a proper prior is
necessary. Our arguments involve a blend of Bayesian and frequentist reasoning,
and are intended to complement the original case made by Gelman (2006) in
support of the folded-t family of priors. First, we generalize the half-Cauchy
prior to the wider class of hypergeometric inverted-beta priors. We derive
expressions for posterior moments and marginal densities when these priors are
used for a top-level normal variance in a Bayesian hierarchical model. We go on
to prove a proposition that, together with the results for moments and
marginals, allows us to characterize the frequentist risk of the Bayes
estimators under all global-shrinkage priors in the class. These theoretical
results, in turn, allow us to study the frequentist properties of the
half-Cauchy prior versus a wide class of alternatives. The half-Cauchy occupies
a sensible 'middle ground' within this class: it performs very well near the
origin, but does not lead to drastic compromises in other parts of the
parameter space. This provides an alternative, classical justification for the
repeated, routine use of this prior. We also consider situations where the
underlying mean vector is sparse, where we argue that the usual conjugate
choice of an inverse-gamma prior is particularly inappropriate, and can lead to
highly distorted posterior inferences. Finally, we briefly summarize some open
issues in the specification of default priors for scale terms in hierarchical
models
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