On the Half-Cauchy Prior for a Global Scale Parameter

Abstract

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.

Prior sensitivity analysis in default Bayesian structural equation modeling.

“…However, many non-conjugate priors have been proposed in the Bayesian literature as more robust (i.e., less influential) alternatives. For example, Gelman (2006) and Polson and Scott (2012) proposed the half-Cauchy prior for random effects variances, which can be implemented in a Gibbs sampler relatively easy through parameter expansion. A second option for random effects variances is a Gamma prior in combination with posterior mode estimates which has been proposed in the context of meta-analysis by Chung, Rabe-Hesketh, and Choi (2013).…”

“…A second option for random effects variances is a Gamma prior in combination with posterior mode estimates which has been proposed in the context of meta-analysis by Chung, Rabe-Hesketh, and Choi (2013). Note that the choice of the prior for residual variances is considerably less important than the prior for the variances of latent variables (e.g., Polson & Scott, 2012) For intercept, mean, and regression parameters a robust alternative is the t-distribution, which has been proposed as prior for logistic models by Gelman, Jakulin, Pittau, and Su (2008) and as error distribution to obtain robust models (e.g., robust growth curve models; Zhang, Lai, Lu, & Tong; 2013). The t-distribution includes the Cauchy distribution as special case when the number of degrees of freedom is set to 1.…”

Sow the Wind, Reap the Whirlwind? Mobilization Cascades in Direct Democratic Initiatives.

“…Using broader priors, for example with mean 0 and standard deviation 1000, would not make a lot of sense, obviously. For the top-level scale parameter σ, I use a half-Cauchy prior [39] with location 0 and scale 2.5.…”

Hierarchical Transmuted Log-Logistic Model: A Subjective Bayesian Analysis

“…The authors suggest to use the half-t family as a prior distribution for variance parameters such the half-Cauchy distribution, that is a special conditionally-conjugate folded-noncentral-t family case of prior distributions for parameters that represent the discrepancy. Even though several studies use the half-Cauchy prior for scale parameter (see for example Polson and Scott 2012), Gelman (2006) used this prior not for scale but for variance parameters and illustrated serious problems with the inverse-Gamma prior which is the most commonly used prior for variance component, see Daniels and Daniels (1998).…”

“…For that, the half-Cauchy prior distribution, cited by several authors such as (Polson and Scott 2012;Gelman 2006), as an alternative prior to a inverse Gamma distribution, was used. Specially, Gelman (2006) made use of this particular prior for variance parameters in hierarchical models which is our case.…”