## Bayesian Analysis

### On the Use of Cauchy Prior Distributions for Bayesian Logistic Regression

#### Abstract

In logistic regression, separation occurs when a linear combination of the predictors can perfectly classify part or all of the observations in the sample, and as a result, finite maximum likelihood estimates of the regression coefficients do not exist. Gelman et al. (2008) recommended independent Cauchy distributions as default priors for the regression coefficients in logistic regression, even in the case of separation, and reported posterior modes in their analyses. As the mean does not exist for the Cauchy prior, a natural question is whether the posterior means of the regression coefficients exist under separation. We prove theorems that provide necessary and sufficient conditions for the existence of posterior means under independent Cauchy priors for the logit link and a general family of link functions, including the probit link. We also study the existence of posterior means under multivariate Cauchy priors. For full Bayesian inference, we develop a Gibbs sampler based on Pólya-Gamma data augmentation to sample from the posterior distribution under independent Student-$t$ priors including Cauchy priors, and provide a companion R package $\mathtt{tglm}$, available at CRAN. We demonstrate empirically that even when the posterior means of the regression coefficients exist under separation, the magnitude of the posterior samples for Cauchy priors may be unusually large, and the corresponding Gibbs sampler shows extremely slow mixing. While alternative algorithms such as the No-U-Turn Sampler (NUTS) in Stan can greatly improve mixing, in order to resolve the issue of extremely heavy tailed posteriors for Cauchy priors under separation, one would need to consider lighter tailed priors such as normal priors or Student-$t$ priors with degrees of freedom larger than one.

#### Article information

Source
Bayesian Anal., Volume 13, Number 2 (2018), 359-383.

Dates
First available in Project Euclid: 7 March 2017

https://projecteuclid.org/euclid.ba/1488855634

Digital Object Identifier
doi:10.1214/17-BA1051

Mathematical Reviews number (MathSciNet)
MR3780427

Zentralblatt MATH identifier
06989952

#### Citation

Ghosh, Joyee; Li, Yingbo; Mitra, Robin. On the Use of Cauchy Prior Distributions for Bayesian Logistic Regression. Bayesian Anal. 13 (2018), no. 2, 359--383. doi:10.1214/17-BA1051. https://projecteuclid.org/euclid.ba/1488855634

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#### Supplemental materials

• Supplementary Material for “On the Use of Cauchy Prior Distributions for Bayesian Logistic Regression”. In the supplementary material, we present additional simulation results for logistic and probit regression with complete separation, along with an appendix that contains the proofs of all theoretical results. The Gibbs sampler developed in the paper can be implemented with the R package tglm, available from the website: https://cran.r-project.org/web/packages/tglm/index.html.