Maximum a Posteriori Estimation in Graphical Models Using Local Linear Approximation
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Sparse structure learning in high-dimensional Gaussian graphical models is an important problem in multivariate statistical signal processing; since the sparsity pattern naturally encodes the conditional independence relationship among variables. However, maximum a posteriori (MAP) estimation is challenging if the prior model admits multiple levels of hierarchy, 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 conditions under which our algorithm is guaranteed to converge to the MAP estimate are explicitly derived and are 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 ℓ_2-norm. Numerical results validate the speed, scalability, and statistical performance of the proposed method.
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