The choice of the design of an experiment, including both selection of design points and choice of sample size, can be viewed as a decision problem. We wish to choose the design which maximises the prior expectation of a utility function which depends on both costs of the experiment and benefits from the information gained. The latter may be realised through a second decision to be made after the experiment. Solving the problem requires the evaluation of this expectation for each candidate design involving summation or integration over all possible outcomes for each design. With non-Gaussian models, where posterior evaluations would typically involve intensive numerical methods such as Markov Chain Monte Carlo, evaluation of the conditional expectation of the utility, given an outcome, becomes computationally demanding and so solving such design problems becomes difficult for all but fairly simple cases.
Bayes linear kinematics (Golstein and Shaw, 2004) offers an alternative approach. It is the Bayes linear analogue of probability kinematics (Jeffrey 1965). It offers a method for propagating changes in belief about some quantities through to others within a Bayes linear structure, for example when the changes result from observing related non-Gaussian variables. We adopt a conjugate relationship between observables and parameters and then update beliefs about other quantities using Bayes linear kinematics.
Applying this approach to the design problem greatly reduces the computational burden and the problem can be solved without the need for intensive numerical methods. The method is illustrated using two examples.