The theory of belief functions provides one way to use mathematical probability in subjective judgment. It is a generalization of the Bayesian theory of subjective probability. When we use the Bayesian theory to quantify judgments about a question, we must assign probabilities to the possible answers to that question. The theory of belief functions is more flexible; it allows us to derive degrees of belief for a question from probabilities for a related question. These degrees of belief may or may not have the mathematical properties of probabilities; how much they differ from probabilities will depend on how closely the two questions are related. Examples of what we would now call belief-function reasoning can be found in the late seventeenth and early eighteenth centuries, well before Bayesian ideas were developed. In 1689, George Hooper gave rules for combining testimony that can be recognized as special cases of Dempster's rule for combining belief functions (Shafer 1986a). Similar rules were formulated by Jakob Bernoulli in his Ars Conjectandi, published posthumously in 1713, and by Johann-Heinrich Lambert in his Neues Organon, published in 1764 (Shafer 1978). Examples of belief-function reasoning can also be found in more recent work, by authors

: Belief functions are mathematical objects defined to satisfy three axioms that look somewhat similar to the Kolmogorov axioms defining probability functions. We argue that there are (at least) two useful and quite different ways of understanding belief functions. The first is as a generalized probability function (which technically corresponds to the inner measure induced by a probability function). The second is as a way of representing evidence. Evidence, in turn, can be understood as a mapping from probability functions to probability functions. It makes sense to think of updating a belief if we think of it as a generalized probability. On the other hand, it makes sense to combine two beliefs (using, say, Dempster's rule of combination) only if we think of the belief functions as representing evidence. Many previous papers have pointed out problems with the belief function approach; the claim of this paper is that these problems can be explained as a consequence of confounding the...

: We introduce a new probabilistic approach to dealing with uncertainty, based on the observation that probability theory does not require that every event be assigned a probability. For a nonmeasurable event (one to which we do not assign a probability), we can talk about only the inner measure and outer measure of the event. In addition to removing the requirement that every event be assigned a probability, our approach circumvents other criticisms of probability-based approaches to uncertainty. For example, the measure of belief in an event turns out to be represented by an interval (defined by the inner and outer measure), rather than by a single number. Further, this approach allows us to assign a belief (inner measure) to an event E without committing to a belief about its negation :E (since the inner measure of an event plus the inner measure of its negation is not necessarily one). Interestingly enough, inner measures induced by probability measures turn out to correspond in a ...

This paper investigates Walley’s concepts of epistemic irrelevance and epistemic independence for imprecise probability models. We study the mathematical properties of irrelevance and independence, and their relation to the graphoid axioms. Examples are given to show that epistemic irrelevance can violate the symmetry, contraction and intersection axioms, that epistemic independence can violate contraction and intersection, and that this accords with informal notions of irrelevance and independence.

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This paper consists of three main parts. First, we give an introduction to Hill’s assumption A (n) and to theory of interval probability, and an overview of recently developed theory and methods for nonparametric predictive inference (NPI), which is based on A (n) and uses interval probability to quantify uncertainty. Thereafter, we illustrate NPI by introducing a variation to the assumption A (n), suitable for inference based on circular data, with applications to several data sets from the literature. This includes attention to comparison of two groups of circular data, and to grouped data. We briefly discuss such inference for multiple future observations. We end the paper with a discussion of NPI and objective Bayesianism.

One of the major criticisms of probabilistic risk assessment is that the requisite input distributions are often not available. Several approaches to this problem have been suggested, including creating a library of standard empirically fitted distributions, employing maximum entropy criteria to synthesize distributions from a priori constraints, and even using `default' inputs such as the triangular distribution. Since empirical information is often sparse, analysts commonly must make assumptions to select the input distributions without empirical justification. This practice diminishes the credibility of the assessment and any decisions based on it. There is no absolute necessity, however, of assuming particular shapes for input distributions in probabilistic risk assessments. It is possible to make the needed calculations using inputs specified only as bounds on probability distributions. We describe such bounds for a variety of circumstances where empirical information is extremely limited, and illustrate how these bounds can be used in computations to represent uncertainty about input distributions far more comprehensively than is possible with current approaches.

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This paper discusses fundamental aspects of inference with imprecise probabilities from the decision theoretic point of view. It is shown why the equivalence of prior risk and posterior loss, well known from classical Bayesian statistics, is no longer valid under imprecise priors. As a consequence, straightforward updating, as suggested by Walley's Generalized Bayes Rule or as usually done in the Robust Bayesian setting, may lead to suboptimal decision functions. As a result, it must be warned that, in the framework of imprecise probabilities, updating and optimal decision making do no longer coincide.