Sparse networks can be found in a wide range of applications, such as biological and communication networks. Inference of such networks from data has been receiving considerable attention lately, mainly driven by the need to understand and control internal working mechanisms. However, while most available methods have been successful at predicting many correct links, they also tend to infer many incorrect links. Precision is the ratio between the number of correctly inferred links and all inferred links, and should ideally be close to 100%. For example, 50% precision means that half of inferred links are incorrect, and there is only a 50% chance of picking a correct one. In contrast, this paper infers links of discrete-time linear networks with very high precision, based on variational Bayesian inference and Gaussian processes. Our method can handle limited datasets, does not require full-state measurements and effectively promotes both system stability and network sparsity. On several of examples, Monte Carlo simulations illustrate that our method consistently has 100% or nearly 100% precision, even in the presence of noise and hidden nodes, outperforming several state-of-the-art methods. The method should be applicable to a wide range of network inference contexts, including biological networks and power systems.