Interval-valued Probability Research Articles

Probabilistic databases for decision analysis

A probabilistic database is defined in a previous article [R. Cavallo and M. Pittarelli, Proc. 13th Int. Conf. on Very Large Databases (VLDB); 1987; see Ref. 9] as a collection of probability distributions over Cartesian products of finite variable domains. the concept is extended here to accommodate interval-valued probabilities. Algebraic operations for both real- and interval-valued probabilities databases analogous to those for relational databases are defined. Techniques for making inferences regarding joint distributions on subsets of the variables over which a probabilistic database is defined are developed. These are illustrated through application to a problem of decision analysis under partial uncertainty. Connections between the probabilistic database formalism and other forms of data representation are discussed.

RECONSTRUCTABILITY ANALYSIS USING PROBABILITY INTERVALS

It is proposed that probability intervals be used in reconstructability analysis. A probability interval is a subinterval of the real interval [0,1]. Regarded as “interval-valued probabilities”, these intervals generalize real-valued probabilities and arise naturally in many situations. They may represent confidence intervals resulting from sampling; imprecisely stated subjective probabilities; known linear equality or inequality constraints; etc. Thus, probability intervals are sometimes a more realistic characterization of uncertainty than are real-valued probabilities. Furthermore, the problem of inconsistency can often be avoided by their use Although the utility of interval-valued probability distributions for the identification problem is emphasized, a reconstruction technique is also developed. This reconstruction method employs a metric distance for interval distributions that is monotonic with respect to model refinement.

Towards a steroid-profiling expert system

Abstract This paper outlines some aspects of a steroid-profiling expert system under development as an alternative to the conventional chemometric approach to metabolic profiling which uses only statistical pattern recognition. Deep knowledge is being structured as a causal model and subjective probabilistic reasoning over the causal model (using interval-valued probabilities) is attempted by means of a graph-theoretic separation criterion for assessing conditional independence. An approach for mapping chromatographic features to corresponding linguistic descriptions is discussed.

Towards a Frequentist Theory of Upper and Lower Probability

We present elements of a frequentist theory of statistics for concepts of upper and lower (interval-valued) probability (IVP), defined on finite event algebras. We consider IID models for unlinked repetitions of experiments described by IVP and suggest several generalizations of standard notions of independence, asymptotic certainty and estimability. Instability of relative freqencies is favoured under our IID models. Moreover, generalizations of Bernoulli's Theorem give some justification for the estimation of an underlying IVP mechanism from fluctuations of relative frequencies. Our results indicate that an objectivist, frequency- or propensity-oriented, view of probability does not necessitate an additive probability concept, and that IVP models can represent a type of indeterminacy not captured by additive probability.

Bayes-like Decision Making with Upper and Lower Probabilities

Abstract We consider the use of interval-valued probabilities to represent the support lent to the hypothesis that the parameter value θ lies in a subset A of the parameter set Θ when we observe x, know the likelihoods {fΘ: θeΘ}, and have some prior information concerning the parameter. Our model for prior information is that of a salient prior distribution in which we have little confidence, although we have much less confidence in any alternative prior. We consider notions of acceptable and coherent decision making as well as notions of being able to achieve a Bayes rule and least commitment. Throughout we are motivated to preserve some of the elements of Bayesian decision making without thereby committing ourselves to unwarranted claims of knowledge.