Abstract
We put forward the Scaled Beta2 (SBeta2) as a flexible and tractable family for modeling scales in both hierarchical and non-hierarchical settings. Various sensible alternatives to the overuse of vague Inverted Gamma priors have been proposed, mainly for hierarchical models. Several of these alternatives are particular cases of the SBeta2 or can be well approximated by it. This family of distributions can be obtained in closed form as a Gamma scale mixture of Gamma distributions, as the Student distribution can be obtained as a Gamma scale mixture of Normal variables. Members of the SBeta2 family arise as intrinsic priors and as divergence based priors in diverse situations, hierarchical and non-hierarchical. The SBeta2 family unifies and generalizes different proposals in the Bayesian literature, and has numerous theoretical and practical advantages: it is flexible, its members can be lighter, as heavy or heavier tailed as the half-Cauchy, and different behaviors at the origin can be modeled. It has the reciprocality property, i.e if the variance parameter is in the family the precision also is. It is easy to simulate from, and can be embedded in a Gibbs sampling scheme. Short of not being conjugate, it is also amazingly tractable: when coupled with a conditional Cauchy prior for locations, the marginal prior for locations can be found explicitly as proportional to known transcendental functions, and for integer values of the hyperparameters an analytical closed form exists. Furthermore, for specific choices of the hyperparameters, the marginal is found to be an explicit “horseshoe prior”, which are known to have excellent theoretical and practical properties. To our knowledge this is the first closed form horseshoe prior obtained. We also show that for certain values of the hyperparameters the mixture of a Normal and a Scaled Beta2 distribution also gives a closed form marginal. Examples include robust normal and binomial hierarchical modeling and meta-analysis, with real and simulated data.
Highlights
The focus of this paper is to propose the Scaled Beta2 (SBeta2) family of distributions, SBeta2(ψ|p, q, b) =Γ(p + q) Γ(p)Γ(q) · b · (( ( ψ b )(p−1) ) + 1)(p+q), for ψ
We show that for certain values of the hyperparameters the mixture of a Normal and a Scaled Beta2 distribution gives a closed form marginal
We justify our proposal of using the SBeta2 family for modeling scales based on a combination of theoretical and practical considerations. It has a natural motivation as a Gamma scaled mixture of a Gamma distribution as shown in Lemma 1. It has the property of reciprocality, i.e., if p(ψ) belongs to the SBeta2 family, so does p(1/ψ), which is not a property of the Gamma/InvertedGamma family
Summary
It has the property of reciprocality, i.e., if p(ψ) belongs to the SBeta family, so does p(1/ψ), which is not a property of the Gamma/InvertedGamma family (the half-Cauchy distribution proposed by Gelman to model standard deviations has this attractive property, and it is reassuring that this is a particular case of our proposal as mentioned before) It is flexible enough for modeling a variety of behaviors at the origin and at the tail, and for specific hyperparameters boundedness at the origin and heavy right tail is obtained, as heavy or even heavier than the Cauchy distribution. C) Again for a conditional Cauchy prior for location, if the square of the scale is modeled as a SBeta, the marginal is no longer a horseshoe prior, but a general closed form result is obtained This strategy leads to a very useful prior distribution, called the Student-SBeta distribution (Fuquene et al, 2014).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
