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Title: In mean and variance variable selection with variational approximations Authors:  Mauro Bernardi - University of Padova (Italy) [presenting]
Luca Maestrini - The Australian National University (Australia)
Giulia Livieri - Scuola Normale Superiore (Italy)
Abstract: Variable selection plays a key role in modern statistical research and learning. Major classes of variable selection approaches are implemented using Markov chain Monte Carlo methods. These methods may be computationally impractical for large-scale problems or complex models and faster approximations are desirable or necessary. We develop an approach to variable selection for heteroscedastic regression models based upon semiparametric mean field variational Bayes. The methodology we propose is suitable for models having linear mean and exponential variance functions with prior specifications that induce sparse solutions on the regression coefficients, namely Bayesian lasso, spike-and-slab, and adaptive spike-and-slab lasso. The use of classic mean field variational Bayes leads to the approximating densities having non-standard forms and challenging numerical problems arise in the determination of the optimal approximation. We achieve tractability by imposing a parametric assumption to the approximate marginal posterior densities of variance regression coefficients. Our iterative optimization of the log-likelihood lower bound includes Newton-type steps with analytical derivative expressions for the parametric component of the approximation. This optimization strategy uses new results that solve recurrent issues of constrained optimization involving multivariate skew-normal variational approximations.