Bayesian design for calibration of physical models

Abstract

We often want to learn about physical processes that are described by complex nonlinear mathematical models implemented as computer simulators. To use a simulator to make predictions about the real physical process, it is necessary to first perform calibration; that is, to use data obtained from a physical experiment to make inference about unknown parameters whilst acknowledging discrepancies between the simulator and reality. The computational expense of many simulators makes calibration challenging. Thus, usually in calibration, we use a computationally cheaper approximation to the simulator, often referred to as an emulator, constructed by fitting a statistical model to the results of a relatively small computer experiment. Although there is a substantial literature on the choice of the design of the computer experiment, the problem of designing the physical experiment in calibration is much less well-studied. This thesis is concerned with methodology for Bayesian optimal designs for the physical experiment when the aim is estimation of the unknown parameters in the simulator.

Optimal Bayesian design for most realistic statistical models, including those incorporating expensive computer simulators, is complicated by the need to numerically approximate an analytically intractable expected utility; for example, the expected gain in Shannon information from the prior to posterior distribution. The standard approximation method is "double-loop" Monte Carlo integration using nested sampling from the prior distribution. Although this method is easy to implement, it produces biased approximations and is computationally expensive. For the Shannon information gain utility, we propose new approximation methods which combine features of importance sampling and Laplace approximations.

These approximations are then used within an optimisation algorithm to find optimal designs for three problems: (i) estimation of the parameters in a nonlinear regression model; (ii) parameter estimation for a misspecified regression model subject to discrepancy; and (iii) estimation of the calibration parameters for a computational expensive simulator. Through examples, we demonstrate the advantages of this combination of methodology over existing methods.

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Identifiers

ORCID for David Woods: orcid.org/0000-0001-7648-429X

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