On the half-cauchy prior for a global scale parameter

By Nicholas G. Polson, James G. Scott. 2011

458 citations recorded by the source corpus.

This paper argues that the half-Cauchy distribution should replace the inverseGamma 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.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/6a1701a9a95a1f42d04c60d80966e9f5a611cfb9

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Citations in the corpus

On Using Bayesian Methods to Address Small Sample Problems (2016, methodology): “Gelman (2006) recommended using half-Cauchy priors for variance components; both analytical work (Polson & Scott, 2012) and simulation work (McNeish & Stapleton, 2016) have shown that this type of prior is superior to other commonly recommended, noninformative priors with small samples.”
Uncertainty Quantification for the Horseshoe (with Discussion) (2016, methodology): “This paper studies the Bayesian approach based on the horseshoe prior [10, 9, 37, 30, 31].”
Posterior Computation with the Gibbs Zig-Zag Sampler (2020, methodology): “Finally, as recommended by Polson and Scott (2012), we choose i.”
Scale-Dependent Priors for Variance Parameters in Structured Additive Distributional Regression (2016, methodology): “As a consequence, several alternatives such as half-normal, half-Cauchy or (proper) uniform priors for the standard deviation have been suggested as default priors in the literature (e.g. Gelman, 2006; Gelman et al., 2008; Polson and Scott, 2012).”
Adaptive posterior contraction rates for the horseshoe (2017, methodology): “In the present paper we study the Bayesian method based on the horseshoe prior [7, 6, 28, 23, 24].”
Bayesian regression tree models for causal inference: regularization, confounding, and heterogeneous effects (2017, unknown): “95), except that we place a half-Cauchy prior over the scale of the leaf parameters with prior median equal to twice the marginal standard deviation of Y (Gelman et al., 2006; Polson et al., 2012).”
Modeling Clustered Data with Very Few Clusters (2016, background): “To date, although analytical arguments for half-t and half-Cauchy have beenmade (e.g., Polson & Scott, 2012), the performance (both absolute and relative to other priors) of these recommendations has not been systematically assessed.”
Bayesian Generalized Additive Models for Location, Scale, and Shape for Zero-Inflated and Overdispersed Count Data (2015, methodology): “Half-Cauchy priors have recently been advocated by Polson and Scott (2012); see also Gelman (2006) for a general discussion of prior specifications in Bayesian hierarchical mixed models.”
The Horseshoe Estimator: Posterior Concentration around Nearly Black Vectors (2014, methodology): “In similar experiments in (Carvalho, Polson and Scott, 2009; Polson and Scott, 2012a) the horseshoe prior outperformed the DE prior, while behaving similarly to a heavy-tailed discrete mixture.”
The Prior Can Often Only Be Understood in the Context of the Likelihood (2017, background): “On the other end, Reference [23] argue persuasively that, from an admissibility point of view, the prior on σ should have heavy tails, eventually advocating the half-Cauchy prior on σ advocated originally by [22].”
Comparison of Inverse Wishart and Separation-Strategy Priors for Bayesian Estimation of Covariance Parameter Matrix in Growth Curve Analysis (2016, methodology): “Because previous studies have suggested that different priors for the variance parameter can be used (e.g., Gelman, 2006; Polson & Scott, 2012), we investigate three priors for marginal variances.”
runjags: An R Package Providing Interface Utilities, Model Templates, Parallel Computing Methods and Additional Distributions for MCMC Models in JAGS (2016, unknown): “…the half-Cauchy distribution has been recommended as a reasonable choice for standard deviation parameters within hierarchical models (Gelman 2006; Polson and Scott 2011), and DuMouchel (1994) gives an argument for the use of π ( τ ) = s 0 ( s 0 + τ ) 2 as a prior for a variance parameter τ in…”
Penalising Model Component Complexity: A Principled, Practical Approach to Constructing Priors (2014, background): “Some known weakly informative priors, like the half-Cauchy distribution on the standard deviation of a normal distribution, can lead to better predictive inference than reference priors (Polson and Scott, 2012).”
Robust meta‐analytic‐predictive priors in clinical trials with historical control information (2014, background): “Spiegelhalter et al. (2004), Gelman (2006), and Polson and Scott (2012) discuss several priors and recommend Half-T distributions, which include the Half-Normal and Half-Cauchy as special cases.”