Belief specification

In general, the level of detail at which we choose to describe our beliefs will depend on

• how interested we are in the various aspects of the problem;
• our ability to specify each aspect of our uncertainty;
• the amount of time and effort that we are willing to expend on the problem;
• how much detail is required from our prior specification in order to extract the necessary information from the data.

We must, therefore, recognise that our analysis depends not only upon the observed data but also upon the level of detail to which we have expressed our beliefs. The formal framework within which we shall express our judgements is as follows.

We begin by supplying an ordered (finite or infinite) list of random quantities, for which we shall make statements of uncertainty. We call the base for our analysis.

For each we specify

1. the expectation, , giving a simple quantification of our belief as to the magnitude of ;
2. the variance, , quantifying our uncertainty or degree of confidence in our judgements of the magnitude of ;
3. the covariance, , expressing a judgement on the relationship between the quantities, quantifying the extent to which observation on may (linearly) influence our belief as to the size of .

These expectations, variances and covariances are specified directly, although this does not preclude us from deducing the values from some larger specification, or even, when this is practical, from a full prior probability distribution. We require that each element of C must have finite prior variance.

For any ordered subcollections, A, B, of elements of C, we write

to denote the variance matrix of the vector of elements of A, and we write

to denote the covariance matrix between the vectors A and B.

We control the level of detail of our investigations by our choice of the collection C. The most detailed collection that we could possibly select would consist of the indicator functions for all of the combinations of possible values of all of the quantities of interest.With this choice of C, we obtain a full probability specification over some underlying outcome space. Sometimes this special case may be appropriate, but for large problems we will usually restrict attention to small subcollections of this collection. (Thus, for example, if there were two quantities Y and Z which we might measure, then C might contain the terms ) It is preferable to work explicitly with the collection of belief specifications that we have actually made rather than to pretend to specify much larger collections of prior belief statements.