Restricted Prior Inference for Complex Uncertainty Structures

Abstract

The problem of representing and analysing partial aspects of uncertainty is examined using a geometric approach. A Hilbert space of random objects is constructed, where the inner product captures aspects of beliefs about the relationship between the objects. Orthogonal direct sums of the Hilbert space are used to restrict the amount of detail that is required for the prior specification. Using minimal assumptions of temporal consistency, this geometric space is adapted to derive the stochastic relationships between the formal restricted partial belief analysis and the corresponding posterior uncertainty judgements.

To illustrate the methodology, a simple multivariate time series dynamic linear model is developed to represent the sales of leading brands of soft-drink from “cash-and-carry” depots. Restricted prior inferences are developed for the pair of variance matrices underlying this model, where uncertainty for a given depot is decomposed into aspects which may be explained with data from that depot, those which may be explained using data from related depots, and those aspects of our uncertainty for our posterior judgements which cannot be explained a priori.