A Bayesian Multilevel Framework for Uncertainty Characterization and the NASA Langley Multidisciplinary UQ Challenge

Variance Decomposition for Statistical Quantities of Interest

“…For this work, the Gaussian kernel was used, in conjunction with a sample size of 5000. Use of the Gaussian reference rule (h 1.06σN −1∕5 , where h is the window width, σ is the sample standard deviation, and N is the sample size) for determination of the window width was found to result in significant oversmoothing of the density, as is also noted in [29]. To identify a more appropriate value of the window width, a combination of least-squares cross validation [30], likelihood cross validation [30], and heuristic visual assessment was used.…”

“…Refer to [16] for a complete solution to subproblem A, including parts A2 and A4. Readers are also referred to [29] for further discussion regarding both theoretical and practical considerations.…”

“…(The observations used for calibration are also shown with vertical lines and markers on the x-axis) Note that accurate characterization of the highly peaked second mode of x 1 requires use of a fairly small window width, resulting in some undersmoothing of the central portion of the distribution. Further discussion of the practical and theoretical aspects of approximating the likelihood function for the Challenge Problem can be found in [29].…”

A unified framework for multilevel uncertainty quantification in Bayesian inverse problems

“…The data ⟨ỹ i ⟩ is then only explained by uncertainty of the forward model inputs as described in Section 2.2, without being subject to prediction errors. Hereafter we will refer this scenario as to involve "perfect" data [59,60].…”

“…Rather than aiming at an unknown constant m, inference concentrates on the hyperparameters θ X that determine the variability of envisaged. An application example where inference targets both parameters of the type m and θ X , in the presence of additional nuisance parameters ⟨ζ i ⟩, can be found in [59,60].…”