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 with a large number of covariates. We introduce a new class of distributions, the Perturbed Unified Skew Normal (PSUN), which generalizes the SUN class and 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 probit and logistic cases. The proposed methodology, based on a straightforward Gibbs sampler algorithm, can be always applied. In particular, in the p > n case, it shows better performances both in terms of mixing and accuracy, compared to the existing methods.