In this paper, we consider the problem of predicting a large scale spatial field using successive noisy measurements obtained by mobile sensing agents. The physical spatial field of interest is discretized and modeled by a Gaussian Markov random field (GMRF) with unknown hyperparameters. From a Bayesian perspective, we design a sequential prediction algorithm to exactly compute the predictive inference of the random field. The prediction algorithm correctly takes into account the uncertainty in hyperparameters in a Bayesian way and also is scalable to be usable for the mobile sensor networks with limited resources. An adaptive sampling strategy is also designed for mobile sensing agents to find the most informative locations in taking future measurements in order to minimize the prediction error and the uncertainty in hyperparameters simultaneously. The effectiveness of the proposed algorithms is illustrated by a numerical experiment.