We consider the general problem of Bayesian binary regression and we introduce a new class of distributions, the Perturbed Unified Skew Normal (pSUN, henceforth), which generalizes the Unified Skew-Normal (SUN) class. We show that the new class is conjugate to any binary regression model, provided that the link function may be expressed as a scale mixture of Gaussian densities. We discuss in detail the popular logit case, and we show that, when a logistic regression model is combined with a Gaussian prior, posterior summaries such as cumulants and normalizing constants can be easily obtained through the use of an importance sampling approach, opening the way to straightforward variable selection procedures. For more general priors, the proposed methodology is based on a simple Gibbs sampler algorithm. We also claim that, in the p > n case, the proposed methodology shows better performances - both in terms of mixing and accuracy - compared to the existing methods. We illustrate the performance through several simulation studies and two data analyses.

翻译：我们考虑贝叶斯二元回归的一般问题，并引入了一类新的分布——扰动统一偏斜正态（以下简称pSUN），它推广了统一偏斜正态（SUN）类。我们证明，只要链接函数可以表示为高斯密度的尺度混合，新类对任何二元回归模型都是共轭的。我们详细讨论了流行的Logit情形，并表明，当逻辑回归模型与高斯先验结合时，可以通过重要性抽样方法轻松获得后验摘要（如累积量和归一化常数），从而为直接变量选择程序开辟道路。对于更一般的先验，所提出的方法基于简单的吉布斯抽样算法。我们还声称，在p>n的情况下，所提出的方法在混合性和准确性方面均优于现有方法。我们通过多项模拟研究和两个数据分析来展示其性能。