Title: Fast Bayesian model selection algorithms for linear regression models
Authors: Mauro Bernardi - University of Padova (Italy)
Claudio Busatto - University of Florence (Italy)
Manuela Cattelan - University of Padova (Italy) [presenting]
Abstract: The issue of model selection for high-dimensional linear regression has been primarily addressed by assuming hierarchical mixtures as prior distributions. A spike component with Dirac probability mass at zero is introduced to exclude irrelevant covariates, thereby leading to Bayesian selection procedures that rely on the computation of the marginal posterior distribution for alternative model configurations. The exploration of the space of competing models is usually performed by means of computationally intensive simulation-based techniques. We address the issue of fast updating the variance-covariance matrix of the posterior distribution and the marginal posterior density itself, after a modification of the current design matrix. First, leveraging a thin QR factorization, novel algorithms to update the posterior variance-covariance matrix are proposed which avoid storage and update the $Q$ matrix thus allowing noticeable savings. Then, the issue of evaluating the marginal posterior is considered, as it represents the bottleneck of any Bayesian model selection procedure. It is shown that the computation of the marginal posterior relies on the inverse of the $R$ matrix, hence we develop a new methodology to update both this inverse and the related marginal posterior after the modification of the current design matrix. These methods do not need computationally intensive inversions of large dimensional matrices when performing marginal posterior evaluations.