For the problem of inferring a Gaussian graphical model (GGM), this work explores the application of a recent approach from the multiple testing literature for graph inference. The main idea of the method by Rebafka et al. (2022) is to model the data by a latent variable model, the so-called noisy stochastic block model (NSBM), and then use the associated ${\ell}$-values to infer the graph. The inferred graph controls the false discovery rate, that means that the proportion of falsely declared edges does not exceed a user-defined nominal level. Here it is shown that any test statistic from the GGM literature can be used as input for the NSBM approach to perform GGM inference. To make the approach feasible in practice, a new, computationally efficient inference algorithm for the NSBM is developed relying on a greedy approach to maximize the integrated complete-data likelihood. Then an extensive numerical study illustrates that the NSBM approach outperforms the state of the art for any of the here considered GGM-test statistics. In particular in sparse settings and on real datasets a significant gain in power is observed.
