We address the problem of Bayesian variable selection for high-dimensional linear regression. We consider a generative model that uses a spike-and-slab like prior distribution obtained by multiplying a deterministic binary vector, which traduces the sparsity of the problem, with a random Gaussian parameter vector. Such a model allows an expectation-maximization algorithm, optimizing a type-II log-likelihood, to be derived. This marginal log-likelihood involves an Occam's razor term, automatically penalizing the complexity, which is used for model selection. Albeit NP-hard, the algorithm we propose can be relaxed in order to infer a family of models. Model selection is eventually performed afterwards based on Occam's razor. We report numerical comparisons between our method, called spinyReg, and the most recent variable selection algorithms, including lasso, adaptive lasso and stability selection. SpinyReg turns out to perform well compared to those algorithms, especially regarding false detection rates.