There are many different ways in which beliefs may be quantified. Most familiar, perhaps, is the Bayesian approach, in which beliefs about all of the uncertain quantities of interest are represented in terms of a joint probability distribution. In practice, the specification of such a joint probability distribution will often be largely arbitrary due to the difficulty that most of us find in thinking meaningfully and consistently in high numbers of dimensions (or even in low numbers of dimensions - indeed even specifying a single probability may be a daunting task if our answer really matters for some purpose).
Full probabilistic specification is unwieldy as a fundamental expression of prior knowledge in that it requires such an extremely large number of statements of prior knowledge, expressing judgements to so fine a level of detail, that usually we have neither the interest nor the ability to make most of these judgements in a meaningful way. To escape from the straitjacket of full probabilistic specification, we suggest an approach which is related in spirit to the Bayesian approach, but is more straightforward to apply.
Suppose, therefore, that we intend to quantify some aspects of our prior judgements. It is reasonable to require that our subsequent analysis should only be based on those aspects of our beliefs which we are both willing and able to specify. Each number that we specify expresses some aspect of our prior knowledge, and as such requires careful consideration. Our concern is to develop a methodology which allows us to specify and analyse relatively small, carefully chosen collections of quantitative judgements about whichever aspects of a problem are within our ability to specify in a meaningful way.
We begin by describing our basic approach to the quantification of belief.