Bayesian Calibration of Computer Models

Abstract

Summary. We consider prediction and uncertainty analysis for systems which are approximated using complex mathematical models. Such models, implemented as computer codes, are often generic in the sense that by a suitable choice of some of the model's input parameters the code can be used to predict the behaviour of the system in a variety of speci®c applications. However, in any speci®c application the values of necessary parameters may be unknown. In this case, physical observations of the system in the speci®c context are used to learn about the unknown parameters. The process of ®tting the model to the observed data by adjusting the parameters is known as calibration. Calibration is typically effected by ad hoc ®tting, and after calibration the model is used, with the ®tted input values, to predict the future behaviour of the system. We present a Bayesian calibration technique which improves on this traditional approach in two respects. First, the predictions allow for all sources of uncertainty, including the remaining uncertainty over the ®tted parameters. Second, they attempt to correct for any inadequacy of the model which is revealed by a discrepancy between the observed data and the model predictions from even the best-®tting parameter values. The method is illustrated by using data from a nuclear radiation release at Tomsk, and from a more complex simulated nuclear accident exercise.

Emulation of Leaf, Canopy and Atmosphere Radiative Transfer Models for Fast Global Sensitivity Analysis

“…It made that existing emulation packages (e.g., BACCO (Bayesian Analysis of Computer Code Outputs) or GEM-SA (Gaussian Emulation Machine for Sensitivity Analysis) [34,72,73]), as they process only one output, are impractical for RTM sensitivity analysis. To cope with such a high degree of multiple-output, we had implemented a PCA dimensionality reduction procedure into ARTMO's Emulator toolbox and linked it with the GSA toolbox.…”

Efficiently Constraining Climate Sensitivity with Ensembles of Paleoclimate Simulations

“…In this paper, we describe and present results from the first fullscale application of this method, in which it has been applied to simultaneously estimating 25 model parameters in a state-of-the-art AGCM coupled to a slab ocean. One critical input into the process is the estimate of uncertainty of the model inadequacy (also called discrepancy) (Kennedy and O'Hagan 2001) which has so far received surprisingly little attention given its important rôle in applications of this type Rougier 2005). Since this term is as yet poorly characterised, we have performed a range of experiments using assumptions that we expect to cover the plausible range of model inadequacy (both doubling and halving our best prior estimate).…”

“…That is, the uncertainty of the observations is much smaller than the uncertainty of the model error itself (where the "model error" is defined as the difference between reality, and the model evaluated at the "best" set of parameters for the problem in question) . The theory and implications of these issues are also explored in more detail in Kennedy and O'Hagan (2001) and Rougier (2005). The practical effect of accounting for model error is to increase the denominator of the cost function for model fitness, thereby broadening the posterior distribution and increasing the spread of the results compared to when a perfect model assumption is made.…”

Bayesian quantification of thermodynamic uncertainties in dense gas flows

“…In the following, we use a likelihood function resulting from a multiplicative/additive error model, similar to that used in [22] for calibrating the closure coefficients of turbulence model from measured velocity profiles in a turbulent boundary layer. We refer to [42], [22] and the references cited therein for more details about possible choices for the construction of likelihood functions.…”

“…(see [42]), where x i and x j are two distinct observation abscissas separated by the length scale 10 α X, X being a reference length scale, here taken equal to the airfoil chord. The coefficients σ and α are supplementary parameters intrinsic to the statistical model.…”