Why Bayes linear methods?

Our approach is characterised by several features:

• It is subjectivist: we deal with uncertain quantities (like the average daily temperature during next year's summer); we hold beliefs about such quantities (we expect it to be cold again); and we can measure related quantities (such as whether it snows on Chrismas day). We combine such information in order to help us revise our beliefs about these uncertain quantities.
• It is based upon expectation as a primitive: we deal with expectations of uncertain quantities, rather than probabilities of events. (If this worries you as seeming rather restrictive, recall that probabilities are expectations of indicator functions: let be any event, and define to be the random quantity which takes the value if occurs, and otherwise. Then .)
• It is practicable: we demand of you only that you specify the information (in the form of expectations, variances and covariances) that you are both willing and able to provide. In contrast, a standard Bayesian treatment typically requires the provision of full multidimensional probability distribution over all the quantities of interest (whether or not this can be done meaningfully).
• It is simple; traditional Bayesian methods are frequently extremely demanding in terms of the computational effort needed for an analysis, whereas the Bayes linear approach preserves via its linearity a natural tractability.
• It is insightful: the information contained within our constructions is readily accessible, and some of the tools that we provide yield insights that are difficult to obtain under other approaches. This follows partly because a traditional Bayesian approach, for example, can involve a mass of specifications (much of it arguably arbitrary) and calculations from which it is difficult to extract revealing summaries.
• It subsumes the traditional Bayesian analysis as a special limiting case (in the sense of expectations over indicator functions).
• The Bayes linear approach can be considered to be an approximation to the full Bayesian treatment in the case of certain limited specifications.
• It is logically supportable. (See [3] for a discussion of underlying philosophical issues.)

Thus, in our view our approach is logically founded; tractable and simple to apply; easily understood and well-defined; genuine and achievable.