Jaynes and find that generalization to an arbitrary number of levels of fidelity and parallelisation becomes rather easy.
We devise the fully Bayesian uncertainty quantification method in a notation following the tradition of E.T. The recent hype about machine learning has long spilled over to computational engineering but fails to acknowledge that machine learning is a big data problem and that, in computational engineering, we usually face a little data problem. By quantifying the uncertainties of the parameters themselves too, we show that “learning” or optimising those parameters has little meaning when data is little and, thus, justify all our mathematical efforts. What is left is a low-dimensional and feasible numerical integral depending on the choice of kernels, thus allowing for a fully Bayesian treatment. This is done analytically for all but the nonlinear or inseparable kernel function parameters. We propagate the parameter uncertainties by averaging the predictions and the prediction uncertainties over all the possible parameters. Thus, we avoid the awkward logical dilemma of having to choose parameters and of neglecting that choice’s uncertainty. Departing from there, we move away from the common non-Bayesian practice of optimization and marginalize the parameters instead. They assumed each level to be describable by a Gaussian process, and used low-fidelity simulations to improve inference on costly high-fidelity simulations. In 2000, Kennedy and O’Hagan proposed a model for uncertainty quantification that combines data of several levels of sophistication, fidelity, quality, or accuracy, e.g., a coarse and a fine mesh in finite-element simulations.
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