Abstract: Gaussian Markov random field is a probabilistic model with multivariate Gaussian distribution and conditional independence assumptions. Gaussian mean field is a basic variational inference method on the Gaussian Markov random field, which computes the lower bound of the objective function through variational transformation with free distribution of the variables factorized into clusters. The structure selection of free distribution plays an important role in variational inference, and it is critical to the tradeoff between the variational accuracy and the computational complexity. This paper deals with the structure selection criterion and algorithm issues for the Gaussian mean field, and then provides a new structure selection criterion and an efficient structure selection algorithm. First, the concepts of coupling and quasi-coupling are proposed to measure the dependence among variable clusters of the Gaussian Markov random field model, and the coupling-accuracy theorem is proved for the Gaussian mean field, which provides the quasi-coupling as the new structure selection criterion. Then a normalized structure selection algorithm is designed based on the quasi-coupling criterion and the normalization technique for Gaussian mean field, which avoids unbalanced computational complexity among clusters through cluster normalization. Finally, numerical comparison experiments are presented to demonstrate the validity and efficiency of the normalized structure selection algorithm.