Bayes linear sufficiency and systems of expert posterior assessments

Data arising in the form of expert assessments are received by a decision maker. The decision maker is required to estimate a set of unknown quantities, and receives expert assessments at varying levels of accuracy, on samples of the quantities of interest. We present a Bayes linear analysis of this problem. In the absence of other assessments, the decision maker will accept as his or her current estimate of any single quantity the most accurate received assessment of that quantity. This leads to a sufficiency property which allows a simple decomposition of the error structure of assessments. Bayes linear estimation is then used by the decision maker to estimate each quantity of interest given an arbitrary collection of received assessments. The analysis is motivated throughout by a practical context in which a large company needs to estimate costs for renovation of assets. The methodology is illustrated with a numerical example.