Maximum a posteriori estimation in graphical models using local linear approximation

Sparse structure learning in high‐dimensional Gaussian graphical models is an important problem in multivariate statistical inference, since the sparsity pattern naturally encodes the conditional independence relationship among variables. However, maximum a posteriori (MAP) estimation is challenging under hierarchical prior models, and traditional numerical optimization routines or expectation–maximization algorithms are difficult to implement. To this end, our contribution is a novel local linear approximation scheme that circumvents this issue using a very simple computational algorithm. Most importantly, the condition under which our algorithm is guaranteed to converge to the MAP estimate is explicitly stated and is shown to cover a broad class of completely monotone priors, including the graphical horseshoe. Further, the resulting MAP estimate is shown to be sparse and consistent in the ‐norm. Numerical results validate the speed, scalability and statistical performance of the proposed method.