This thesis deals with the development and application of numerical
integration techniques for use in Bayesian Statistics. In particular,
it describes how imbedded sequences of positive interpolatory
integration rules (PIIR's) obtained from Gauss-Hermite product rules
can extend the applicability and efficiency of currently available
numerical methods.
The numerical strategy suggested by Naylor and Smith (1982) is
reviewed, criticised and applied to some examples with real and
artificial data. The performance of this strategy is assessed from
the viewpoint of 3 criteria: reliability, efficiency and accuracy.
The imbedded sequences of PIIR’s are introduced as an alternative and
an extension to the above strategy for two major reasons. Firstly,
they provide a rich class of spatially ditributed rules which are
particularly useful in high dimensions. Secondly, they provide a way
of producing more efficient integration strategies by enabling
approximations to be updated sequentially through the addition of new
nodes at each step rather than through changing to a completely new
set of nodes.
Finally, the Improvement in the reliability and efficiency achieved by
the adaption of an integration strategy based on PIIR's is
demonstrated with various illustrative examples. Moreover, it is
directly compared with the Gibbs sampling approach introduced recently
by Gelfand and Smith (1988).University of Sheffiel
