Geometric Foundations for Interval-Based Probabilities

Annals of Mathematics and Artificial Intelligence, 1998

The need to reason with imprecise probabilities arises in a wealth of situations ranging from pooling of knowledge from multiple experts to abstraction-based probabilistic planning. Researchers have typically represented imprecise probabilities using intervals and have developed a wide array of di erent techniques to suit their particular requirements. In this paper we provide an analysis of some of the central issues in representing and reasoning with interval probabilities. At the focus of our analysis is the probability cross-product operator and its interval generalization, the cc-operator. We perform an extensive study of these operators relative to manipulation of sets of probability distributtions. This study provides insight into the sources of the strengths and weaknesses of various approaches to handling probability intervals. We demonstrate the application of our results to the problems of inference in interval Bayesian networks and projection and evaluation of abstract probabilistic plans.

2014

In many engineering situations, we need to make decisions under uncertainty. In some cases, we know the probabilities p i of different situations i; these probabilities should add up to 1. In other cases, we only have expert estimates of the degree of possibility µ i of different situations; in accordance with the possibility theories, the largest of these degrees should be equal to 1. In practice, we often only know these degrees p i and µ i with uncertainty. Usually, we know the upper bound and the lower bound on each of these values. In other words, instead of the exact value of each degree, we only know the interval of its possible values, so we need to process such interval-valued degrees. Before we start processing, it is important to find out which values from these intervals are actually possible. For example, if only have two alternatives, and the probability of the first one is 0.5, then-even if the original interval for the second probability is wide-the only possible value of the second probability is 0.5. Once the intervals are narrowed down to possible values, we need to compute the range of possible values of the corresponding characteristics (mean, variance, conditional probabilities and possibilities, etc.). For each such characteristic, first, we need to come up with an algorithm for computing its range. In many engineering applications, we have a large amount of data to process, and many relevant decisions need to be made in real time. Because of this, it is important to make sure that the algorithms for computing the desired ranges are as fast as possible. We present expressions for narrowing interval-valued probabilities and possibilities and for computing characteristics such as mean, conditional probabilities, and conditional possibilities. A straightforward computation of these expressions would take time which is quadratic in the number of inputs n. We show that in many cases, linear-time algorithms are possible-and that no algorithm for computing these expressions can be faster than linear-time.

Annals of Mathematics and Artificial Intelligence, 2009

Two approaches to logic programming with probabilities emerged over time: Bayesian reasoning and probabilistic satisfiability (PSAT). The attractiveness of the former is in tying the logic programming research to the body of work on Bayes networks. The second approach ties, from the point of view of computation, reasoning about probabilities to linear programming, and allows for natural expression of imprecision in probabilities via the use of intervals. In this paper we construct precise semantics for one PSAT-based formalism for reasoning with interval probabilities: disjunctive probabilistic logic programs (dp-programs). It has two origins: (1) disjunctive logic programs, a powerful language for knowledge representation, first proposed by Minker in the early eighties (Minker 1982) and (2) a logic programming language with interval probabilities originally considered by Ng and Subrahmanian (Inform Comput 101(2):150–201, 1993; J Autom Reason 10(2):191–235, 1993). We show that the probability ranges of atoms and formulas in dp-programs cannot be expressed as single intervals. We construct the precise description of the set of models for the class of dp-programs and study the computational complexity of this problem, as well as the problem of consistency of a dp-program. We also study the conditions under which our semantics coincides with the single-interval semantics originally proposed by Ng and Subrahmanian. Our work sheds light on the complexity of constructing reasoning formalisms for imprecise probabilities and suggests that interval probabilities alone are inadequate to support such reasoning.

2004

In many real-life situations, we are interested in the value of a physical quantity y that is difficult or impossible to measure directly. To estimate y, we find some easier-to-measure quantities x1, .

The Florida AI Research Society, 2015

Possibilistic logic is an important framework for representing and reasoning with uncertain and inconsistent pieces of information. Standard possibilistic logic expressions are propositional logic formulas associated with positive real degrees belonging to [0,1]. Recently, a flexible representation of uncertain information, where the weights associated with formulas or possible worlds are in the form of intervals, has been proposed. This paper focuses on the problem of normalization of intervalbased possibility distributions. We provide a natural procedure to normalize a sub-normalized interval-based possibility distribution. This procedure is based on the concept of normalized compatible and standard possibility distributions.

Uncertainty in Artificial Intelligence, 1986

This essay tries to expound a conception of interval measures that pennits a particular approach to partial ignorance decision problems. l'he \ irtue of this approach for artificial reasoning systems is mat me follo wing questions become moot: 1. which secondary crircnon ro apply after maximi1.i ng expected utility. and �how much indeterminacy to represent. The cost of the approach is the need for explicit epistemological foundations: for instance, a role ofacceptana with a parameter that allows various attitudes toward error. Note that epistemological foundations are already desirable for independent reasons. The development is as follows: L. probab ility intervals are useful and natural in A.l. svstems: 2. wide intervals avoid error, but are useless in some risk-sensitive decision-making; 3. yet one may obtain narrower, or otherwise decisive intervals with a more relaxed attitude toward error: 4. if bodies of knowledge can be ordered by their attitude to error, one should perfonn the decision analysis with the acceptable body of knowledge that allows the least error, of those that are useful. The resulting behavior differs from that of a Bayesian probabilist because in the proposal. 5. intervals based on .;; uccessive bodies of l<nowledge are not always nested: 6. the use of a probability for a particular decision does not require commitment to the probability for credence: and 7. there may be no accepcahte body of knowledge that is useful: hence. sometimes no decision is mandated.

Possibilistic logic is a well-known framework for dealing with uncertainty and reasoning under inconsistent knowledge bases. Standard possibilistic logic expressions are propositional logic formulas associated with positive real degrees belonging to [0,1]. However, in practice it may be difficult for an expert to provide exact degrees associated with formulas of a knowledge base. This paper proposes a flexible representation of uncertain information where the weights associated with formulas are in the form of intervals. We first study a framework for reasoning with interval-based possibilistic knowledge bases by extending main concepts of possibilistic logic such as the ones of necessity and possibility measures. We then provide a characterization of an interval-based possibilistic logic base by means of a concept of compatible standard possibilistic logic bases. We show that intervalbased possibilistic logic extends possibilistic logic in the case where all intervals are singletons. Lastly, we provide computational complexity results of deriving plausible conclusions from interval-based possibilistic bases and we show that the flexibility in representing uncertain information is handled without extra computational costs.

2002

Conditionalization, i.e., computation of a conditional probability distribution given a joint probability distribution of two or more random variables is an important operation in some probabilistic database models. While the computation of the conditional probability distribution is straightforward when the exact point probabilities are involved, it is often the case that such exact point probability distributions of random variables are not known, but are known to lie in a particular interval.

… , 2008. FUZZ-IEEE …, 2008

Many AI researchers argue that probability theory is only capable of dealing with uncertainty in situations where a fully specified joint probability distribution is available, and conclude that it is not suitable for application in AI systems. Probability intervals, however, constitute a means for expressing incompleteness of information. We present a method for computing probability interval! for probabilities of interest from a partial specification of a joint probability distribution. Our method improves on earlier approaches by all owing for independency relation ships between statistical variables to be exploited .