Elicitation of prior distributions for a multivariate normal distribution
This thesis focuses on elicitation methods for quantifying an expert's subjective opinion about a multivariate normal distribution. Firstly, it is assumed that the expert's opinion can be adequately represented by a natural conjugate prior distribution (a normal inverse-Wishart distribution) and an elicitation method is developed in which the expert performs various assessment tasks that enable the hyperparameters of the distribution to be estimated. An example illustrating use of the method is given. There are some choices in the way hyperparameters are determined and empirical work underlies the choices made. The empirical work aimed to provide a basis for choosing between alternative assessment tasks that may be used in the elicitation method and to examine different ways of using the elicited assessments to estimate the hyperparameters of the prior distribution. In particular, we compare two methods for estimating a spread matrix. The method is implemented in an interactive computer program that questions the expert and forms the subjective distribution. In some practical situations, it may not be possible to accurately represent an expert's opinions by a natural conjugate prior distribution, especially as the conjugate prior description suffers from some restrictions in the manner it represents dependencies between the mean vector and the covariance matrix. As a more flexible alternative, non-conjugate prior distributions are considered in which independent prior distributions for the mean vector and spread matrix are employed. A method of eliciting a prior distribution for the mean when it is assumed to be a multivariate normal distribution is developed. The implementation of the method is given through a pilot study. The prior distribution for the variance is assumed to have one of two forms: either an inverse-Wishart distribution or a generalised inverse-Wishart distribution. An elicitation method is developed for each of these forms of prior distribution. An example illustrating the implementation of the methods is given. Finally, the elicitation methods for the conjugate and the non-conjugate prior distributions are studied and compared in depth through an experiment with subject-matter experts. In this experiment two assessment tasks are used: one is related to the distribution of a sample mean and the other to the distribution of an individual item. A comparison is made between the expert assessments for these two types of task and marked differences are observed.