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This paper proposes a novel uncertainty quantification framework for computationally demanding systems characterized by a large vector of non-Gaussian uncertainties. It combines state-of-the-art techniques in advanced Monte Carlo sampling with Bayesian formulations. The key departure from existing works is the use of inexpensive, approximate computational models in a rigorous manner. Such models can readily be derived by coarsening the discretization size in the solution of the governing PDEs, increasing the time step when integration of ODEs is performed, using fewer iterations if a nonlinear solver is employed or making use of lower-order models. It is shown that even in cases where the inaccurate models provide very poor approximations of the high-fidelity response, statistics of the latter can be quantified accurately with significant reductions in the computational effort. Multiple approximate models can be used, and rigorous confidence bounds of the estimates produced are provided at all stages.