Fast Bayesian variable selection for high dimensional linear models: marginal solo spike and slab priors
The authors consider high-dimensional linear regression models where the number $p$ of parameters is much greater than the number $n$ of observations. To include the idea of sparsity it is assumed that the long vector of regression parameters has many zero components. The problem is to select the non-zero components, i.e. the essential regressors. This is modeled by a binary latent vector $Z$ of the same dimension as the regression parameter which indicates whether a regression coefficient is ``in or out''. The prior distribution of an ``out'' coefficient is usually a point mass at 0 (spike prior) and that of an ``in'' coefficient a flat distribution (slab prior). The posteriori distribution of $Z$ is used to select the model with the highest posteriori probability. Such Bayesian variable selection procedures rely on Markov Chain Monte Carlo (MCMC) methods. One drawback is the feasibility of MCMC in the high-dimensional case. The authors propose their marginal solo spike and slab priors. They use a particular model involving the spike and slab priors one after another for one regression coefficient only, and a conjugate Gaussian prior for the other coefficients. Only $p$ probabilities must be calculated which can be done very fast and parallel. No MCMC is needed. Under mild conditions strong selection consistency can be proved. A simulation study shows that the new approach provides essentially equivalent results compared with the usual (expensive) spike and slab MCMC method.
scientific article; zbMATH DE number 7021706
scientific article