Methodological and Computational Advances for High–Dimensional Bayesian Regression with Binary and Categorical Responses

Probit and logistic regressions are among the most popular and well-established formulations to model binary observations, thanks to their plain structure and high interpretability. Despite their simplicity, their use poses non-trivial hindrances to the inferential procedure, particularly from a computational perspective and in high-dimensional scenarios. This still motivates thriving active research for probit, logit, and a number of their generalizations, especially within the Bayesian community. Conjugacy results for standard probit regression under normal and unified skew-normal (SUN) priors appeared only recently in the literature. Such findings were rapidly extended to different generalizations of probit regression, including multinomial probit, dynamic multivariate probit and skewed Gaussian processes among others. Nonetheless, these recent developments focus on specific subclasses of models, which can all be regarded as instances of a potentially broader family of formulations, that rely on partially or fully discretized Gaussian latent utilities. As such, we develop a unified comprehensive framework that encompasses all the above constructions and many others, such as tobit regression and its extensions, for which conjugacy results are yet missing. We show that the SUN family of distribution is conjugate for all models within the broad class considered, which notably encompasses all formulations relying on likelihoods given by the product of multivariate Gaussian densities and cumulative distributions, evaluated at a linear combination of the parameter of interest. Such a unifying framework is practically and conceptually useful for studying general theoretical properties and developing future extensions. This includes new avenues for improved posterior inference exploiting i.i.d. samplers from the exact SUN posteriors and recent accurate and scalable variational Bayes (VB) approximations and expectation-propagation, for which we derive a novel efficient implementation. Along a parallel research line, we focus on binary regression under logit mapping, for which computations in high dimensions still pose open challenges. To overcome such difficulties, several contributions focus on solving iteratively a series of surrogate problems, entailing the sequential refinement of tangent lower bounds for the logistic log-likelihoods. For instance, tractable quadratic minorizers can be exploited to obtain maximum likelihood (ML) and maximum a posteriori estimates via minorize-maximize and expectation-maximization schemes, with desirable convergence guarantees. Likewise, quadratic surrogates can be used to construct Gaussian approximations of the posterior distribution in mean-field VB routines, which might however suffer from low accuracy in high dimensions. This issue can be mitigated by resorting to more flexible but involved piece-wise quadratic bounds, that however are typically defined in an implicit way and entail reduced tractability as the number of pieces increases. For this reason, we derive a novel tangent minorizer for logistic log-likelihoods, that combines the quadratic term with a single piece-wise linear contribution per each observation, proportional to the absolute value of the corresponding linear predictor. The proposed bound is guaranteed to improve the accuracy over the sharpest among quadratic minorizers, while minimizing the reduction in tractability compared to general piece-wise quadratic bounds. As opposed to the latter, its explicit analytical expression allows to simplify computations by exploiting a renowned scale-mixture representation of Laplace random variables. We investigate the benefit of the proposed methodology both in the context of penalized ML estimation, where it leads to a faster convergence rate of the optimization procedure, and of VB approximation, as the resulting accuracy improvement over mean-field strategies can be substantial in skewed and high-dimensional scenarios.