We present a probabilistic learning inference that assimilates data (target set) into a parameterized large stochastic computational model resulting from discretizing a stochastic boundary value problem (BVP). A target is imposed on a vector-valued random quantity of interest (QoI), observed as the stochastic solution of the BVP. The probabilistic inference estimates the posterior probability model, which is constrained both by the second-order moment of the random residue of the BVP stochastic equations and the target set composed of statistical moments of the QoI. We assume that evaluating a single realization of the BVP is computationally expensive, so the training dataset comprises only a few points differing from big data approaches. The presented application contributes to three-dimensional stochastic homogenization of heterogeneous linear elastic media, specifically when the mesoscale and macroscale are not separated.