In Gaussian graphical models, the likelihood equations must typically be solved iteratively. We investigate two algorithms: A version of iterative proportional scaling which avoids inversion of large matrices, and an algorithm based on convex duality and operating on the covariance matrix by neighbourhood coordinate descent, corresponding to the graphical lasso with zero penalty. For large, sparse graphs, the iterative proportional scaling algorithm appears feasible and has simple convergence properties. The algorithm based on neighbourhood coordinate descent is extremely fast and less dependent on sparsity, but needs a positive definite starting value to converge. We give an algorithm for finding such a starting value for graphs with low colouring number. As a consequence, we also obtain a simplified proof for existence of the maximum likelihood estimator in such cases.

翻译：在高斯图模型中，似然方程通常需通过迭代求解。我们研究两种算法：一种避免大矩阵求逆的迭代比例缩放版本，以及一种基于凸对偶性、通过邻域坐标下降法作用于协方差矩阵的算法（对应零惩罚的图形套索）。对于大规模稀疏图，迭代比例缩放算法具有可行性与简单收敛特性。基于邻域坐标下降的算法计算极快且对稀疏性依赖较小，但需要正定初始值以保证收敛。我们提出一种针对低着色数图的初始值求解算法，并由此给出此类情形下极大似然估计存在性的简化证明。