I. Introdustion

The fundamental reality that motivates interest in any form of probability is variability. There is variability over instances, over space, and over time. As a consequence, we experience uncertainty: a lack of surety in the face of variability. Although uncertain, we intuitively recognize that some things are more uncertain than others; variability itself is variable. That uncertainty has degrees also supports our attempts at measurement. Thus, the overarching definition of subjective probability is that it is such a measure, one that captures a degree of belief, a degree of judged certainty. Surrounding this general notion are several major variant approaches to subjective probability.

A first approach is to treat subjective probabilities as an interpretation of mathematical probability theory. Thus, Section II of this chapter will begin with probability theory as a mathematical system and derive subjective probability as an interpretation of a correspondence between this theory and reality. Subjective probabilities, being personal, need to come from the person. There are two general variants for assessing these degrees of belief: the direct expression of likelihoods and the inference of likelihoods from expressed preferences. Section III derives these variants from an axiomatic system for a qualitative probability structure designed to connect probability more directly to the underlying behavior it is intended to represent. In Section IV, a broader argument is presented as a second general approach to subjective probability. The claim is that subjective probability is not just one of many interpretations; but, rather, subjectivity is an inherent aspect of all uses of probability, i.e. that all probability statements are subjective. Bayesian inference as an outgrowth of this approach is described. Section V describes a third general approach to subjective probability, one which disconnects from traditional mathematical probability theory. Treated as a measure of a degree of belief or of subjective uncertainty, it is noted that uncertainty has not always been seen as a unitary construct. Specifically, from the early days of probability theory, a distinction has been claimed between uncertainty arising from the weight of evidence and uncertainty arising from the balance of evidence. Dempster-Shafer degrees of belief are outlined to exemplify meaningful measures of uncertainty that are distinct from probability theory.