Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper)

By Andrew Gelman. 2004

3,839 citations recorded by the source corpus.

Various noninformative prior distributions have been suggested for scale parameters in hierarchical models. We construct a new folded-noncentral-t family of conditionally conjugate priors for hierarchical standard deviation pa- rameters, and then consider noninformative and weakly informative priors in this family. We use an example to illustrate serious problems with the inverse-gamma family of "noninformative" prior distributions. We suggest instead to use a uni- form prior on the hierarchical standard deviation, using the half-t family when the number of groups is small and in other settings where a weakly informative prior is desired. We also illustrate the use of the half-t family for hierarchical modeling of multiple variance parameters such as arise in the analysis of variance.

Subfields: Statistics and Probability

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Citations in the corpus

A Bayesian model for repeated measures zero-inflated count data with application to outpatient psychiatric service use (2010, methodology): “However, this choice can be deceptively informative because it concentrates most of the prior mass near zero, which, as Gelman (2006) notes, can lead to unstable posterior estimates since the posterior becomes improper as → 0.”
Efficient and Scalable Bayesian Neural Nets with Rank-1 Factors (2020, background): “Weakly informative priors such as the Cauchy are often preferred for robustness as they concentrate less probability at the mean thanks to heavier tails (Gelman et al., 2006).”
A weakly informative default prior distribution for logistic and other regression models (2008, methodology): “Following Gelman and Pardoe (2007), we distinguish between regression inputs and predictors.”
Deep Evidential Regression (2019, background): “A large focus within Bayesian inference is on placing prior distributions over hierarchical models to estimate uncertainty [12, 13].”
High dimensional structured additive regression models: Bayesian regularization, smoothing and predictive performance (2011, background): “Note, however, that a flat prior p.δl/∝1, which was recommended by Gelman (2006), can be written as an improper inverse gamma prior as well with al =−0:5 and bl =0.”
Moving beyond noninformative priors: why and how to choose weakly informative priors in Bayesian analyses (2019, methodology): “An noninformative prior on σB with large ν yields a standard mixed-effects model with random parameters (Gelman 2006).”
How Many Countries for Multilevel Modeling? A Comparison of Frequentist and Bayesian Approaches (2013, methodology): “More sophisticated priors have been proposed (Natarajan and Kass 2000; Gelman 2006); however, I employ specifications which are commonly used by researchers (e.g. Spiegelhalter et al. 1997; Gelman and Hill 2007).”
A re-evaluation of random-effects meta-analysis (2009, background): “It has been suggested that a prior distribution on τ might comprise the positive part of a symmetric distribution around 0, say Cauchy or normal (Gelman, 2006).”
Sparse Variational Analysis of Linear Mixed Models for Large Data Sets. (2011, background): “Although the parameterization is redundant, it has been noted that the redundant parameterizations are often useful for computational reasons and for inducing new classes of priors with appealing properties (Gelman, 2006).”
A Bayesian Analysis of Body Mass Index Data From Small Domains Under Nonignorable Nonresponse and Selection (2010, background): “(20) Finally, a priori, we assume that β and σ 2 are independent with β ∼ Normal(θ0, 0) and (21) p(σ 2) = 1/(1 + σ 2)2, σ 2 > 0, where θ0 and 0 are specified as done by Nandram and Choi (2005), thereby avoiding the ill-behaved prior distribution σ−2 ∼ Gamma(a,a) with a = 0.001 (see Gelman 2006).”
TensorFlow Distributions (2017, methodology): “)) The half-Cauchy and half-Student t distributions are often used as “weakly informative” priors, which exhibit heavy tails, for variance parameters in hierarchical models [16].”