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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. The originality of the work is to consider inference through relaxing the model and using a type-II log-likelihood maximization based on an EM algorithm. Model selection is performed afterwards relying on Occam’s razor and on a path of models found by the EM algorithm. Numerical comparisons between our method, called spinyReg, and state-of-the-art high-dimensional variable selection algorithms (such as lasso, adaptive lasso, stability selection or spike-and-slab procedures) are reported. Competitive variable selection results and predictive performances are achieved on both simulated and real benchmark data sets. An original regression data set involving the prediction of the number of visitors of the Orsay museum in Paris using bike-sharing system data is also introduced, illustrating the efficiency of the proposed approach. The R package spinyReg implementing the method proposed in this paper is available on CRAN. [ABSTRACT FROM AUTHOR]