In this paper, we study data-aided sensing (DAS) for a system consisting of a base station (BS) and a number of nodes, where the BS becomes a receiver that collects measurements or data sets from the nodes that are distributed over a cell. DAS is an iterative data collection scheme that allows the BS to efficiently estimate a target signal (i.e., all nodes' measurements) with a small number of measurements (compared to random polling). In DAS, a set of nodes are selected in each round based on the data sets that are already available at the BS from previous rounds for efficient data collection. We consider DAS for measurements that are correlated Gaussian in this paper. The resulting DAS is referred to as Gaussian DAS. Using the mean squared error (MSE) criterion, in each round, the BS is able to choose a node that has a data set to minimize the MSE of the next round. Furthermore, we generalize Gaussian DAS in two different ways: i) with multiple parallel channels to upload measurements from nodes using random access; ii) with a model selection, where a multi-armed bandit problem formulation is used to combine the model selection with DAS.