When we reduce the number of aspects of uncertainty about which specifications are to be made, we may also simplify the nature of the specification process, by using methods which lead directly to the particular quantifications that we require. For this purpose, we make direct assessments for our (subjective) expectations for the various uncertainties of interest.
The idea of treating expectation as a primitive quantity and specifying
expectation directly rather than through some intermediary probabilistic
specification has been developed at length by various authors. The most
detailed exposition of this approach is described in de Finetti ([1, 2]).
De Finetti uses the term prevision for an expectation elicited directly
and suggests various operational definitions for directly elicited
expectations.
In this formulation, the probability
of an event is simply the expectation or prevision for the associated indicator
function.
We shall therefore assume, in what follows, that we have made various prior expectation statements, through direct elicitation. We cannot give formal rules for specifying prior expectations any more than we can give such rules for specifying prior probabilities in a standard Bayes analysis. Each expectation expresses a subjective choice that must be made given our assessment of the situation in question. Our account concerns the various methods by which we can improve our quantifications of belief, given such initial judgements and relevant data. Thus, while the forming of sensible prior judgements is of fundamental importance, it falls outside the strict remit of this account. We will discuss in a separate report the issues involved in eliciting such restricted prior specifications. All that we shall observe here is that, because any full probability specification over some outcome space is logically equivalent to a specification of the expectation for every random quantity which could possibly be constructed over that outcome space, it must be a substantially easier task to make a careful prior specification of the expectations only for a small subset of such quantities.