Bayes Linear Kinematics and Bayes Linear Bayes Graphical Models

Probability kinematics (Jeffrey, 1965, 1983) furnishes a method for revising a prior probability specification based upon new probabilities over a partition. We develop a corresponding Bayes linear kinematic for a Bayes linear analysis given information which changes our beliefs about a random vector in some generalised way. We derive necessary and sufficient conditions for commutativity of successive Bayes linear kinematics which depend upon the eigenstructure of the Joint kinematic resolution transform. As an application we introduce the Bayes linear Bayes graphical model, which is a mixture of fully Bayesian and Bayes linear graphical models, combining the simplicity of Gaussian graphical models with the ability to allow conditioning on marginal distributions of any form, and exploit Bayes linear kinematics to embed full conditional updates within Bayes linear belief adjustments. The theory is illustrated with a treatment of partition testing for software reliability.