The role of skewed distributions in Bayesian inference: conjugacy, scalable approximations and asymptotics
A broad class of regression models that routinely appear in several fields of application can be expressed as partially or fully discretized Gaussian linear regressions. Besides incorporating the classical Gaussian response setting, this class crucially encompasses probit, multinomial probit and tobit models, among others, and further includes popular extensions of such formulations to multivariate, non-linear and dynamic contexts. The relevance of these representations has motivated decades of active research within the Bayesian field. A main reason for this constant interest is that, unlike for the Gaussian response setting, the posterior distributions induced by these models do not seem to belong to a known and tractable class, under the commonly-assumed Gaussian priors. This has led to the development of several alternative solutions for posterior inference relying either on sampling-based methods or on deterministic approximations, that often experience scalability, mixing and accuracy issues, especially in high dimension. In this seminar, I will review, unify and extend recent advances in Bayesian inference and computation for such a class of models, proving that unified skew-normal (SUN) distributions (which include Gaussians as a special case) are conjugate to the general form of the likelihood induced by these formulations. This result opens new avenues for improved posterior inference, under a broad class of widely-implemented models, via novel closed-form expressions, tractable Monte Carlo methods based on i.i.d. samples from the exact SUN posterior, and more accurate and scalable approximations from variational Bayes and expectation-propagation. These results will be further extended, in asymptotic regimes, to the whole class of Bayesian parametric models via novel limiting approximations relying on generalized skew-normal distributions.