Compositional models and conditional independence in evidence theory

The goal of the paper is twofold. The first is to show that some of the ideas for representation of multidimensional distributions in probability and possibility theories can be transferred into evidence theory. Namely, we show that multidimensional basic assignments can be rather efficiently represented in a form of so-called compositional models. These models are based on the iterative application of the operator of composition, whose definition for basic assignments as well as its properties are presented. We also prove that the operator of composition in evidence theory is in a sense generalization of its probabilistic counterpart.The second goal of the paper is to introduce a new definition of conditional independence in evidence theory and to show in what sense it is superior to that formerly introduced by other authors.     

On various ways of tackling incomplete information in statistics

This short paper discusses the contributions made to the featured section on Low Quality Data. We further refine the distinction between the ontic and epistemic views of imprecise data in statistics. We also question the extent to which likelihood functions can be viewed as belief functions. Finally we comment on the data disambiguation effect of learning methods, relating it to data reconciliation problems.     

Independence for full conditional probabilities: Structure,factorization, non-uniqueness,and Bayesian networks

This paper examines concepts of independence for full conditional probabilities; that is, for set-functions that encode conditional probabilities as primary objects, and that allow conditioning on events of probability zero. Full conditional probabilities have been used in economics, in philosophy, in statistics, in artificial intelligence. This paper characterizes the structure of full conditional probabilities under various concepts of independence; limitations of existing concepts are examined with respect to the theory of Bayesian networks. The concept of layer independence (factorization across layers) is introduced; this seems to be the first concept of independence for full conditional probabilities that satisfies the graphoid properties of Symmetry, Redundancy, Decomposition, Weak Union, and Contraction. A theory of Bayesian networks is proposed where full conditional probabilities are encoded using infinitesimals, with a brief discussion of hyperreal full conditional probabilities.     

Rejoinder on “Statistical reasoning with set-valued information: Ontic vs. epistemic views”

This note replies to comments made on our contribution to the Low Quality Data debate.     

Independence relations for exponential fields

We give four different independence relations on any exponential field. Each is a canonical independence relation on a suitable Abstract Elementary Class of exponential fields, showing that two of these are NSOP₁-like and non-simple, a third is stable, and the fourth is the quasiminimal pregeometry of Zilber's exponential fields, previously known to be stable (and uncountably categorical). We also characterise the fourth independence relation in terms of the third, strong independence.     

The reduction and fusion of fuzzy covering systems based on the evidence theory

This paper studies reduction of a fuzzy covering and fusion of multi-fuzzy covering systems based on the evidence theory and rough set theory. A novel pair of belief and plausibility functions is defined by employing a method of non-classical probability model and the approximation operators of a fuzzy covering. Then we study the reduction of a fuzzy covering based on the functions we presented. In the case of multiple information sources, we present a method of information fusion for multi-fuzzy covering systems, by which objects can be well classified in a fuzzy covering decision system. Finally, by using the method of maximum flow, we discuss under what conditions, fuzzy covering approximation operators can be induced by a fuzzy belief structure.     

Transitive sets in Euclidean Ramsey theory

A finite set X in some Euclidean space Rn is called Ramsey if for any k there is a d such that whenever Rd is k-coloured it contains a monochromatic set congruent to X. This notion was introduced by Erd?s, Graham, Montgomery, Rothschild, Spencer and Straus, who asked if a set is Ramsey if and only if it is spherical, meaning that it lies on the surface of a sphere. This question (made into a conjecture by Graham) has dominated subsequent work in Euclidean Ramsey theory.In this paper we introduce a new conjecture regarding which sets are Ramsey; this is the first ever ‘rival’ conjecture to the conjecture above. Calling a finite set transitive if its symmetry group acts transitively—in other words, if all points of the set look the same—our conjecture is that the Ramsey sets are precisely the transitive sets, together with their subsets. One appealing feature of this conjecture is that it reduces (in one direction) to a purely combinatorial statement. We give this statement as well as several other related conjectures. We also prove the first non-trivial cases of the statement.Curiously, it is far from obvious that our new conjecture is genuinely different from the old. We show that they are indeed different by proving that not every spherical set embeds in a transitive set. This result may be of independent interest.     

Independence and atoms

We prove that if the range of a probability measure contains an interval , 0$">, then there are infinitely many (nontrivial) independent events in this probability space.

     

Schreier sets in Ramsey theory

We show that Ramsey theory, a domain presently conceived to guarantee the existence of large homogeneous sets for partitions on -tuples of words (for every natural number ) over a finite alphabet, can be extended to one for partitions on Schreier-type sets of words (of every countable ordinal). Indeed, we establish an extension of the partition theorem of Carlson about words and of the (more general) partition theorem of Furstenberg-Katznelson about combinatorial subspaces of the set of words (generated from -tuples of words for any fixed natural number ) into a partition theorem about combinatorial subspaces (generated from Schreier-type sets of words of order any fixed countable ordinal). Furthermore, as a result we obtain a strengthening of Carlson's infinitary Nash-Williams type (and Ellentuck type) partition theorem about infinite sequences of variable words into a theorem, in which an infinite sequence of variable words and a binary partition of all the finite sequences of words, one of whose components is, in addition, a tree, are assumed, concluding that all the Schreier-type finite reductions of an infinite reduction of the given sequence have a behavior determined by the Cantor-Bendixson ordinal index of the tree-component of the partition, falling in the tree-component above that index and in its complement below it.

     

Discounted combination of unreliable evidence using degree of disagreement

When Dempster’s rule is used to implement a combination of evidence, all sources are considered equally reliable. However, in many real applications, all the sources of evidence may not have the same reliability. To resolve this problem, a number of methods for discounting unreliable sources of evidence have been proposed in which the estimation of the discounting (weighting) factors is crucial, especially when prior knowledge is unavailable. In this paper, we propose a new degree of disagreement through which discounting factors can be generated for discounting combinations of unreliable evidence. The new degree of disagreement is established using distance of evidence. It can be experimentally verified that our degree of disagreement describes the disagreements or differences among bodies of evidence well and that it can be effectively used in discounting combinations of unreliable evidence.     

Basic concepts for a theory of evaluation: Hierarchical aggregation via autodistributive connectives in fuzzy set theory

The ‘aggregative operator’ introduced by Combi (1982) in his analysis of ‘basic concepts for a theory of evaluation’ in this journal is based on the axioms of associativity, continuity, and monotonicity. In this paper, an alternative approach is given where associativity is replaced by another axiom, viz. autodistributivity, implying a kind of hierarchical aggregation instead of the former horizontal one. The new set of properties yields a general class of operators for connecting fuzzy sets which are formally similar to the aggregative operators, but are purely compensative (in a strong sense). Moreover, this modified approach generalizes a recent approach based on generalized means as compensative connectives. It appears to be similar to expected utility approaches in decision making under uncertainty, and seems to be a very promising tool to handle representation problems as they grow from other fields of human decision making, e.g. of multicriteria analysis and of welfare theory.     

Independence polynomials of circulants with an application to music

The independence polynomial of a graph G is the generating function I(G,x)=∑_k≥0i_kx^k, where i_k is the number of independent sets of cardinality k in G. We show that the problem of evaluating the independence polynomial of a graph at any fixed non-zero number is intractable, even when restricted to circulants. We provide a formula for the independence polynomial of a certain family of circulants, and its complement. As an application, we derive a formula for the number of chords in an n-tet musical system (one where the ratio of frequencies in a semitone is 2^1/n) without ‘close’ pitch classes.     

Building a binary outranking relation in uncertain, imprecise and multi-experts contexts: The application of evidence theory

We consider multicriteria decision problems where the actions are evaluated on a set of ordinal criteria. The evaluation of each alternative with respect to each criterion may be uncertain and/or imprecise and is provided by one or several experts. We model this evaluation as a basic belief assignment (BBA). In order to compare the different pairs of alternatives according to each criterion, the concept of first belief dominance is proposed. Additionally, criteria weights are also expressed by means of a BBA. A model inspired by ELECTRE I is developed and illustrated by a pedagogical example.     

On the Continuity and Absolute Continuity of Random Closed Sets

Abstract

In the case of real-valued random variables, the concept of absolute continuity is well-defined in terms of the absolute continuity of the probability law of a random variable with respect to the usual Lebesgue measure, since both are acting on the same Borel sigma algebra on the real line. Naturally, the same extends to random vectors with real components. A satisfactory and commonly accepted definition of absolute continuity of random closed sets is not available, while in various applications this would help in clarifying the kind of randomness of a random set. We introduce here a definition that is shown to be an extension of the concept related to real-valued random variables, such that also for random sets it is true that absolute continuity implies continuity. Significant examples and counter examples are presented to illustrate the role of our definition in concrete cases. The relationship between our definition and others in well-accepted literature is shown.     

Longest Cycles in 3-connected Graphs with Given Independence Number

Fouquet and Jolivet conjectured that a k-connected graph of order n and independence number α ≥ k has a cycle of length at least [Fouquet and Jolivet, Problèmes combinatoires et théorie des graphes Orsay (1976), Problems, page 438]. Here we prove this conjecture for k=3.     

Random sets as imprecise random variables

Given a random set coming from the imprecise observation of a random variable, we study how to model the information about the probability distribution of this random variable. Specifically, we investigate whether the information given by the upper and lower probabilities induced by the random set is equivalent to the one given by the class of the probabilities induced by the measurable selections; together with sufficient conditions for this, we also give examples showing that they are not equivalent in all cases.     

Cardinal admissibility and optimability in finite nonarchimedean decision theory

Well-known equivalences between nondominatedness or admissibility of an act and its Bayes optimality with respect to some prior probability measure are generalized to the linear nonarchimedean setting in which utilities are lexicographically ordered vectors and state probabilities are matrices that must commute multiplicatively with the matrix probabilities used in defining mixed acts to test for potential dominance. The principal result, that an act can either be made optimal or is dominated by some mixed act, but not both, is tantamount to a theorem of the alternative for doubly indexed, lexicographically ordered vectors.The authors appreciate the careful reading and valuable expositional advice from Liping Liu, two referees, and Associate Editor Harold P. Benson. For the first author, support from the Freeman Professorship for Doctoral Studies and Research is gratefully acknowledged.