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Two of the most significant challenges in uncertainty quantification pertain to the high computational cost for simulating complex physical models and the high dimension of the random inputs. In applications of practical interest, both of these problems are encountered, and standard methods either fail or are not feasible. To overcome the current limitations, we present a generalized formulation of a Bayesian multi-fidelity Monte-Carlo (BMFMC) framework that can exploit lower-fidelity model versions in a small data regime. The goal of our analysis is an efficient and accurate estimation of the complete probabilistic response for high-fidelity models. BMFMC circumvents the curse of dimensionality by learning the relationship between the outputs of a reference high-fidelity model and potentially several lower-fidelity models. While the continuous formulation is mathematically exact and independent of the low-fidelity model's accuracy, we address challenges associated with the small data regime (i.e., only a small number of 50 to 300 high-fidelity model runs can be performed). Specifically, we complement the formulation with a set of informative input features at no extra cost. Despite the inaccurate and noisy information that some low-fidelity models provide, we demonstrate that accurate and certifiable estimates for the quantities of interest can be obtained for uncertainty quantification problems in high stochastic dimensions, with significantly fewer high-fidelity model runs than state-of-the-art methods for uncertainty quantification. We illustrate our approach by applying it to challenging numerical examples such as Navier-Stokes flow simulations and fluid-structure interaction problems.