The theory of interval-probability as a unifying concept for uncertainty

AbstractThe concept of interval-probability is motivated by the goal to generalize classical-probability so that it can be used for describing uncertainty in general. The foundations of the theory are based on a system of three axioms — in addition to Kolmogorov's axioms — and definitions of independence as well as of conditional-probability. The resulting theory does not depend upon interpretations of the probability concept. As an example of generalising classical results Bayes' theorem is described — other theorems are only mentioned

PROBABILISTIC INFERENCE FOR INTERVAL PROBABILITIES IN DECISION-MAKING PROCESSES

The present paper considers one approach to Bayes’ formula based probabilistic inference under interval values of relevant probabilities; the necessity of it is caused by the impossibility to obtain reliable deterministic values of the required probabilistic evaluations. The paper shows that the approach proves to be the best from the viewpoint of the required amount of calculations and visual representation of the results. The execution of the algorithm of probabilistic inference is illustrated using a classical task of decision making related to oil mining. For visualisation purposes, the state of initial and target information is modelled using probability trees.

A nonparametric predictive alternative to the Imprecise Dirichlet Model: the case of a known number of categories

Nonparametric Predictive Inference (NPI) is a general methodology to learn from data in the absence of prior knowledge and without adding unjustified assumptions. This paper develops NPI for multinomial data where the total number of possible categories for the data is known. We present the general upper and lower probabilities and several of their properties. We also comment on differences between this NPI approach and corresponding inferences based on Walley's Imprecise Dirichlet Model

Imprecise Probability and Chance

Understanding probabilities as something other than point values (e.g., as intervals) has often been motivated by the need to find more realistic models for degree of belief, and in particular the idea that degree of belief should have an objective basis in “statistical knowledge of the world.” I offer here another motivation growing out of efforts to understand how chance evolves as a function of time. If the world is “chancy” in that there are non-trivial, objective, physical probabilities at the macro-level, then the chance of an event e that happens at a given time is e goes to one continuously or not is left open. Discontinuities in such chance trajectories can have surprising and troubling consequences for probabilistic analyses of causation and accounts of how events occur in time. This, coupled with the compelling evidence for quantum discontinuities in chance’s evolution, gives rise to a “(dis)continuity bind” with respect to chance probability trajectories. I argue that a viable option for circumventing the (dis)continuity bind is to understand the probabilities “imprecisely,” that is, as intervals rather than point values. I then develop and motivate an alternative kind of continuity appropriate for interval-valued chance probability trajectories

Nonparametric predictive inference for comparison of two diagnostic tests

An important aim in diagnostic medical research is comparison of the accuracy of two diagnostic tests. In this paper, comparison of two diagnostic tests is presented using nonparametric predictive inference (NPI) for future order statistics. The tests are assumed to be applied on the same individuals from two groups, e.g., healthy and diseased individuals, or from three groups with a known ordering, e.g., adding a group of severely diseased individuals to the two group scenario. Our comparison is explicitly in terms of lower and upper probabilities for proportions of correctly diagnosed future individuals from each group, for a given total number of such individuals. We include in our comparison the possibility that it is more important to get a correct diagnosis for individuals from one group than from another group