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Bayesian binary regression is a prosperous area of research due to thecomputational challenges encountered by currently available methods either forhigh-dimensional settings or large datasets, or both. In the present work, wefocus on the expectation propagation (EP) approximation of the posteriordistribution in Bayesian probit regression under a multivariate Gaussian priordistribution. Adapting more general derivations in Anceschi et al. (2023), weshow how to leverage results on the extended multivariate skew-normaldistribution to derive an efficient implementation of the EP routine having aper-iteration cost that scales linearly in the number of covariates. This makesEP computationally feasible also in challenging high-dimensional settings, asshown in a detailed simulation study.

Efficient expectation propagation for posterior approximation in high-dimensional probit models