Graphical Bayesian models in multivariate expert judgements and conditional external Bayesianity
This thesis addresses the multivariate version of the group decision problem (French, 1985), where the opinions about the possible values of n random variables in a problem, expressed as subjective conditional probability density functions of the k members of a group of experts, are to be combined together into a single probability density. A particular type of graphical chain model a bit more general than an influence diagram defined as a partially complete chain graph (PCG) is used to describe the multivariate causal (ordered) structure of associations between those n random variables. It is assumed that the group has a commonly agreed. PCG but the members diverge about the actual conditional probability densities for the component variables in the common PCG. From this particular situation we investigate some suitable solutions. The axiomatic approach to the group decision problem suggests that the group adopts a combination algorithm which demands, at least on learning information which is common to the members and which preserves the originally agreed PCG structure, that the pools of conditional densities associated with the PCG are externally Bayesian (Madansky, 1964). We propose a logarithmic characterisation for such conditionally externally Bayesian (CEB) poolings which is more flexible than the logarithmic characterisation proposed by Genest et al. (1986). It is illustrated why such a generalisation is practically quite useful allowing, for example, the weights attributed to the joint probability assessments of different individuals in the pool to differ across the distinct conditional probability densities which compound each joint density. A major advantage of this scheme is that it may allow the weights given to the group's members to vary according to the areas of prediction they can perform best. It is also shown that the group's commitment to being CEB on chain elements can be accomplished with the group appearing externally Bayesian on the whole PCG. Another feature of the CEB logarithmic pools is that with them the impossibility theorems related to the preservation of independence by opinion pools can be avoided. Yet, in the context of the axiomatic approach, we show the conditions under which the types of pools that satisfy McConway's (1981) marginalization property, i.e. the linear pools, can also be CEB. Also, the expert judgement problem (French, 1985) is investigated through the Bayesian modelling approach where a supra-Bayesian decision maker treats the experts' opinions as data in the usual Bayesian framework. Graphical representations of standard combination models are discussed in the light of the issues of dependence among experts and sufficiency of experts' statements in certain cases. Most importantly, a supra-Bayesian analysis of uncalibrated experts allows the establishment of a link between the axiomatic and Bayesian modelling approaches. Reconciliation rules which are externally Bayesian are obtained. This result most naturally extends those rules to be CEB in the above mentioned multivariate structures.