We describe a general approach to the comparison of two stochastic specifications over a collection of random quantities and then extend the comparison to collections of stochastic specifications. This comparison derives from the eigenstructure of the belief transform, which we construct in full generality for partially specified belief structures. We describe an application of the methodology, namely the comparison of hypotheses. Given various competing probabilistic specifications for a collection of observable quantities, we use the belief transform to separate those quantities for which the different models make very different predictions (and therefore which may be used to distinguish between the models) from those quantities for which the different models make similar predictions (and therefore which may be used to assess the suitability of the general class of models under consideration). Finally, we describe a particular application of hypothesis comparison which relates to the modelling of a collection of time series derived from an aluminum smelting process.
"Belief Transforms and the Comparison of Hypotheses." Ann. Statist. 19 (4) 2067 - 2089, December, 1991. https://doi.org/10.1214/aos/1176348386