B ayes Linear Analysis
Michael Goldstein · Wiley StatsRef: Statistics Reference Online · 2015
Abstract The Bayes linear approach is concerned with problems in which we want to combine prior judgments of uncertainty with observational data, and we use expected value rather than probability as the primitive for expressing these judgments. This distinction is of particular relevance in problems with too many sources of information for us to be comfortable in making a meaningful full joint prior probability specification of the type required for Bayesian inference. For such problems, the Bayes linear approach is similar in spirit to a full Bayesian analysis but offers a practical methodology for analyzing partially specified beliefs for large problems. We describe the adjustment of prior means and variances given data and explain the various interpretations that we may attribute to such adjustments. We describe the interpretive and diagnostic cycle for the analysis: (i) we interpret the expected effects of the adjustment, a priori; (ii) given observations, we interpret the outcome of the actual adjustment; (iii) we make diagnostic comparisons between observed and expected beliefs. The approach to modeling and analysis through the second‐order exchangeability representation theorem is described, and various real world applications are discussed.