Abstract
Traditionally Bayesian decision-theoretic design of experiments proceeds by choosing a design to minimise expectation of a given loss function over the space of all designs. The loss function encapsulates the aim of the experiment, and the expectation is taken with respect to the joint distribution of all unknown quantities implied by the statistical model that will be fitted to observed responses. In this paper, an extended framework is proposed whereby the expectation of the loss is taken with respect to a joint distribution implied by an alternative statistical model. Motivation for this includes promoting robustness, ensuring computational feasibility and for allowing realistic prior specification when deriving a design. To aid in exploring the new framework, an asymptotic approximation to the expected loss under an alternative model is derived, and the properties of different loss functions are established. The framework is then demonstrated on a linear regression versus full-treatment model scenario, on estimating parameters of a non-linear model under model discrepancy and a cubic spline model under an unknown number of basis functions.
Highlights
The Bayesian decision-theoretic approach (Chaloner and Verdinelli, 1995) is a natural framework to plan experiments in many fields of science and engineering
A Bayesian design minimises the expectation of the loss over the space of all possible designs where expectation is with respect to a probability distribution over all unknown quantities implied by the statistical model that will be fitted upon observation of the experimental responses
An extended framework is proposed for Bayesian design whereby the expectation of the loss is taken with respect to the probability distribution implied by an alternative model to the one being fitted to the observed responses
Summary
The Bayesian decision-theoretic approach (Chaloner and Verdinelli, 1995) is a natural framework to plan experiments in many fields of science and engineering It starts with specification of a loss function representing the aim of the experiment. A Bayesian design minimises the expectation of the loss over the space of all possible designs where expectation is with respect to a probability distribution over all unknown quantities implied by the statistical model that will be fitted upon observation of the experimental responses. We propose an extended framework for Bayesian decision-theoretic design of experiments This is achieved by defining the expected loss by taking expectation with respect to a probability distribution implied by an alternative statistical model (termed a designer model). The Supplementary Material includes proofs and derivations of the results, and the cubic spline model example
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