Introduction

Bayes linear methodology provides a quantitative structure for expressing our beliefs and systematic methods for revising these beliefs given observational data. Particular emphasis is placed upon interpretive and diagnostic features of the analysis. The approach is similar in spirit to the standard Bayes analysis, but is constructed so as to avoid much of the burden of specification and computation of the full Bayes case. From a foundational view, the Bayes analysis emerges as a special case of the Bayes linear approach. From a practical view, Bayes linear methods offer a way of tackling problems which are too complex to be handled by standard Bayesian tools.

This report is the first of a series describing Bayes linear methods. In this document, we introduce some of the basic machinery of the theory. Examples, computational issues, detailed derivations of results and methods for belief elicitation will be addressed in related reports. In particular, [9] contains a simple tutorial guide to the material in this report, by means of a simple example, with details as to how the relevant calculations may be programmed in the computer language [B/D].

We cover the following material.

Section 2

concerns our basic approach to quantifying uncertainty and details the specification requirements for the Bayes linear analysis.

Section 3

defines and interprets the notions of adjusted expectation and adjusted variance for a collection of quantities, and explains the role of canonical directions in summarising the effects of an adjustment.

Section 4

concerns the types of diagnostic comparisons that we may make after we have evaluated the belief adjustment. In particular, we discuss the role of the bearing of the adjustment in summarising the overall magnitude and nature of the changes between prior and adjusted beliefs.

Section 5

covers the role of partial adjustments for analysing beliefs which are modified in stages.

Section 6

Bayes linear methods are so named as, formally, they derive their properties from the linear structure of inner product spaces rather than the boolean structure of probability spaces. This section summarises the geometry underlying the adjustment of beliefs.