  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.
