Interpretation

How should we interpret adjusted expectations? There are four inter-related interpretations that we can offer.

• The simplest interpretation is to view the quantity as an `estimator' of the value of X, which combines the data with simple aspects of our prior beliefs in an intuitively plausible manner and which leads to a useful methodology. Alternately, if we have extensive data sources to draw upon, then we may construct our prior judgements from these sources and use our approach to develop `estimators' which can be viewed as complementary to certain standard estimators in multivariate analysis.
• The second interpretation is to view adjusted expectation simply as a primitive which quantifies certain aspects of our beliefs, in a similar manner to the original expectation statement. Indeed, in de Finetti's formal development of prevision, the principle operational definition that he offers is that our prevision for X is the value x which we would choose if we were forced to suffer a penalty

   

  where k is a constant defining the units of loss. In this view adjusted expectation simply expresses the extension of our choice of preferences from the certain choice x to the random choice

   

  In the special case where the elements are the indicator functions for a partition, then this is equivalent to de Finetti's choice for the operational definition of the conditional prevision, , of X given each event . Under this view, adjusted expectations are simply informative summaries, generalising the corresponding conditional expectations defined over indicator functions.

• If we are committed in principle to a full Bayes view based on complete probabilistic specification of all uncertainties, then we may view adjusted expectations as offering simple tractable approximations to a full Bayes analysis for complicated problems. In addition, the various interpretive measures and diagnostic tests which we shall develop below offer insights which are relevant to any full Bayes analysis.
• We have described three alternative views of adjusted expectation, each of which has merit in certain contexts and reflects the contrasting views that may be held concerning the revision of beliefs. However, we hold a fourth view, which, by proceeding directly by foundational arguments, subsumes each of the above views. This view explains why we should view adjusted expectation as a primitive, the precise sense in which adjusted expectation may be viewed as an `estimator', and the general properties which may be claimed for the estimate. Further, it reverses our third interpretation above by identifying a full Bayes analysis as a simple special case of the general analysis which we advocate.

Our immediate intention is to describe the practical machinery of our approach. Therefore, we do not at this point intend to take logical and philosophical diversions into foundational issues, and we shall develop the formal relationship between belief adjustment and belief revision elsewhere. Instead, for now we will move between the first three interpretations that we have listed above, viewing adjusted expectation as an intuitively plausible numerical summary statement about our beliefs given the data. There is no implication that this value will fully express our genuine revised belief concerning the expectation of X. Rather, we have been explicit as to precisely which aspects of our prior beliefs have been utilised in order to assess the adjusted expectation. As with any other formal analysis that we might carry out, adjusted expectations offer logical information in quantitative form which we may use as we deem appropriate to improve our actual posterior judgements.