Laplace approximation for Bayesian variable selection via Le Cam's one-step procedure

Relevant feature selection in high-dimensional settings is a central challenge in modern scientific research and decision-making. While many existing methods offer strong statistical guarantees, they are often computationally intractable in high-dimensional problems. To address this issue, we introduce a novel Laplace approximation method based on Le Cam’s one-step procedure, termed \textsf{OLAP}. This approach is specifically designed to alleviate computational burdens while maintaining statistical rigor. Under standard high-dimensional assumptions, we establish that \textsf{OLAP} achieves consistent variable selection. Moreover, the method yields a posterior distribution that can be efficiently explored in polynomial time via a simple Gibbs sampling algorithm. We demonstrate the effectiveness of OLAP through applications to logistic and Poisson regression models, using both simulated and real data.