An extension of the Unified Skew-Normal family of distributions and application to Bayesian binary regression

We consider the general problem of Bayesian binary regression and we introduce a new class of distributions, the Perturbed Unified Skew Normal (pSUN), which generalizes the 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 constant, can be easily obtained, opening the way to straightforward variable selection procedures. For more general priors, the proposed methodology is based on a straightforward Gibbs sampler algorithm. We also claim that, in the p > n case, it shows better performances both in terms of mixing and accuracy, compared to the existing methods. We illustrate the performance of the proposal through a simulation study and two real datasets, one covering the standard case with n >> p and the other related to the p >> n situation.