An Extension of the Unified Skew-Normal Family of Distributions and its Application to Bayesian Binary Regression

We consider the Bayesian binary regression model and we introduce a new class of distributions, the Perturbed Unified Skew-Normal ((Formula presented.), henceforth), which generalizes the Unified Skew-Normal ((Formula presented.)) 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 CDFs. 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 easily be obtained through the use of an importance sampling approach, opening the way to straightforward variable selection procedures. For more general prior distributions, the proposed methodology is based on a simple Gibbs sampler algorithm. We also claim that, in the (Formula presented.) case, our proposal presents 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. Supplementary materials for this article, including the R package (Formula presented.), are available online.