标题： 关于在贝叶斯逻辑回归中使用柯西先验分布的应用 显示英文标题

标题： On the Use of Cauchy Prior Distributions for Bayesian Logistic Regression

摘要： 在逻辑回归中，当预测变量的线性组合可以完美分类样本中的部分或全部观测值时，就会发生分离现象，结果导致回归系数的最大似然估计有限不存在。Gelman等（2008）建议在逻辑回归中使用独立的柯西分布作为回归系数的默认先验分布，即使在分离的情况下也是如此，并在他们的分析中报告了后验众数。由于柯西先验的均值不存在，一个自然的问题是，在分离情况下，回归系数的后验均值是否存在。我们证明了定理，这些定理为在logit链接和包括probit链接在内的广义链接函数族下，独立柯西先验的后验均值的存在性提供了必要且充分的条件。我们还研究了多变量柯西先验下的后验均值的存在性。对于完整的贝叶斯推断，我们开发了一个基于Polya-Gamma数据增强的吉布斯采样器，用于从独立的学生t先验包括柯西先验下的后验分布中进行抽样，并在补充材料中提供了一个配套的R包。我们通过实证表明，即使在分离情况下回归系数的后验均值存在，柯西先验的后验样本的幅度可能异常大，相应的吉布斯采样器表现出极慢的混合速度。虽然像Stan中的No-U-Turn采样器这样的替代算法可以显著改善混合效果，但为了解决分离情况下柯西先验的极端重尾后验问题，需要考虑尾部较轻的先验，如正态先验或自由度大于1的学生t先验。 显示英文摘要

摘要： 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 Polya-Gamma data augmentation to sample from the posterior distribution under independent Student-t priors including Cauchy priors, and provide a companion R package in the supplement. 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 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.