Faria, Alvaro Eduardo (1996) Graphical Bayesian models in multivariate expert judgements and conditional external Bayesianity. PhD thesis, University of Warwick.

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Abstract

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.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Bayesian statistical decision theory, Multivariate analysis -- Graphic methods
Official Date:	October 1996
Dates:	Date Event October 1996 Submitted
Institution:	University of Warwick
Theses Department:	Department of Statistics
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Smith, J. Q., 1953-
Sponsors:	Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)
Extent:	vi, 133 leaves
Language:	eng

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