Modelling of Uncertainty and Bi–Variable Maps (original) (raw)

Bell-Type Inequalities for Bivariate Maps on Orthomodular Lattices

Foundations of Physics, 2015

Bell-type inequalities on orthomodular lattices, in which conjunctions of propositions are not modeled by meets but by maps for simultaneous measurements (s-maps), are studied. It is shown, that the most simple of these inequalities, that involves only two propositions, is always satisfied, contrary to what happens in the case of traditional version of this inequality in which conjunctions of propositions are modeled by meets. Equivalence of various Bell-type inequalities formulated with the aid of bivariate maps on orthomodular lattices is studied. Our investigations shed new light on the interpretation of various multivariate maps defined on orthomodular lattices already studied in the literature. The paper is concluded by showing the possibility of using s-maps and j-maps to represent counterfactual conjunctions and disjunctions of non-compatible propositions about quantum systems.

Uncertainty Modeling by Bilattice-Based Squares and Triangles

IEEE Transactions on Fuzzy Systems, 2000

In this paper, Ginsberg's/Fitting's theory of bilattices, and in particular the associated constructs of bilatticebased squares and triangles, is introduced as an attractive framework for the representation of uncertain and potentially conflicting information, paralleling Goguen's L-fuzzy set theory. We recall some of the advantages of bilattice-based frameworks for handling fuzzy sets and systems, provide the related structures with adequately defined graded versions of the basic logical connectives, and study their properties and relationships.

Unifying practical uncertainty representations–I: Generalized p-boxes

International Journal of Approximate Reasoning, 2008

There exist several simple representations of uncertainty that are easier to handle than more general ones. Among them are random sets, possibility distributions, probability intervals, and more recently Ferson's p-boxes and Neumaier's clouds. Both for theoretical and practical considerations, it is very useful to know whether one representation is equivalent to or can be approximated by other ones. In this paper, we define a generalized form of usual p-boxes. These generalized p-boxes have interesting connections with other previously known representations. In particular, we show that they are equivalent to pairs of possibility distributions, and that they are special kinds of random sets. They are also the missing link between p-boxes and clouds, which are the topic of the second part of this study.

On Lattice Structure of the Probability Functions on L

2012

In this paper, the set of all probability functions on L * is studied, where L * is the lattice of bothvalued fuzzy sets or intuitionistic fuzzy sets. It is shown that the set of all probability functions on L * endowed with two appropriate operations has a monoid structure which is also a distributive complete lattice. Also the lattice structure of the set of all probability functions on L * induced by an appropriate function on [0, 1] to itself is studied. Some lattice (dual) isomorphisms are discussed that suggests probabilities on L * could be considered in the framework of theories modeling imprecision.

Uncertainty measures: The big picture

ArXiv, 2021

Probability theory is far from being the most general mathematical theory of uncertainty. A number of arguments point at its inability to describe second-order (‘Knightian’) uncertainty. In response, a wide array of theories of uncertainty have been proposed, many of them generalisations of classical probability. As we show here, such frameworks can be organised into clusters sharing a common rationale, exhibit complex links, and are characterised by different levels of generality. Our goal is a critical appraisal of the current landscape in uncertainty theory.

Representing uncertainty on set-valued variables using belief functions

Artificial Intelligence, 2010

A formalism is proposed for representing uncertain information on set-valued variables using the formalism of belief functions. A set-valued variable X on a domain Ω is a variable taking zero, one or several values in Ω. While defining mass functions on the frame 2 2 Ω is usually not feasible because of the double-exponential complexity involved, we propose an approach based on a definition of a restricted family of subsets of 2 Ω that is closed under intersection and has a lattice structure. Using recent results about belief functions on lattices, we show that most notions from Dempster-Shafer theory can be transposed to that particular lattice, making it possible to express rich knowledge about X with only limited additional complexity as compared to the single-valued case. An application to multi-label classification (in which each learning instance can belong to several classes simultaneously) is demonstrated.

Nearly-Linear uncertainty measures

International Journal of Approximate Reasoning, 2019

Several easy to understand and computationally tractable imprecise probability models, like the Pari-Mutuel model, are derived from a given probability measure P0. In this paper we investigate a family of such models, called Nearly-Linear (NL). They generalise a number of well-known models, while preserving a simple mathematical structure. In fact, they are linear affine transformations of P0 as long as the transformation returns a value in [0, 1]. We study the properties of NL measures that are (at least) capacities, and show that they can be partitioned into three major subfamilies. We investigate their consistency, which ranges from 2-coherence, the minimal condition satisfied by all, to coherence, and the kind of beliefs they can represent. There is a variety of different situations that NL models can incorporate, from generalisations of the Pari-Mutuel model, the ε-contamination model and other models to conflicting attitudes of an agent towards low/high P0-probability events (both prudential and imprudent at the same time), or to symmetry judgments. The consistency properties vary with the beliefs represented, but not strictly: some conflicting and partly irrational moods may be compatible with coherence. In a final part, we compare NL models with their closest, but only partly overlapping, models, neo-additive capacities and probability intervals.

The Geometry of Uncertainty - The Geometry of Imprecise Probabilities

Artificial Intelligence: Foundations, Theory, and Algorithms, Springer Nature, 2011

The principal aim of this book is to introduce to the widest possible audience an original view of belief calculus and uncertainty theory. In this geometric approach to uncertainty, uncertainty measures can be seen as points of a suitably complex geometric space, and manipulated in that space, for example, combined or conditioned. In the chapters in Part I, Theories of Uncertainty, the author offers an extensive recapitulation of the state of the art in the mathematics of uncertainty. This part of the book contains the most comprehensive summary to date of the whole of belief theory, with Chap. 4 outlining for the first time, and in a logical order, all the steps of the reasoning chain associated with modelling uncertainty using belief functions, in an attempt to provide a self-contained manual for the working scientist. In addition, the book proposes in Chap. 5 what is possibly the most detailed compendium available of all theories of uncertainty. Part II, The Geometry of Uncertainty, is the core of this book, as it introduces the author’s own geometric approach to uncertainty theory, starting with the geometry of belief functions: Chap. 7 studies the geometry of the space of belief functions, or belief space, both in terms of a simplex and in terms of its recursive bundle structure; Chap. 8 extends the analysis to Dempster’s rule of combination, introducing the notion of a conditional subspace and outlining a simple geometric construction for Dempster’s sum; Chap. 9 delves into the combinatorial properties of plausibility and commonality functions, as equivalent representations of the evidence carried by a belief function; then Chap. 10 starts extending the applicability of the geometric approach to other uncertainty measures, focusing in particular on possibility measures (consonant belief functions) and the related notion of a consistent belief function. The chapters in Part III, Geometric Interplays, are concerned with the interplay of uncertainty measures of different kinds, and the geometry of their relationship, with a particular focus on the approximation problem. Part IV, Geometric Reasoning, examines the application of the geometric approach to the various elements of the reasoning chain illustrated in Chap. 4, in particular conditioning and decision making. Part V concludes the book by outlining a future, complete statistical theory of random sets, future extensions of the geometric approach, and identifying high-impact applications to climate change, machine learning and artificial intelligence. The book is suitable for researchers in artificial intelligence, statistics, and applied science engaged with theories of uncertainty. The book is supported with the most comprehensive bibliography on belief and uncertainty theory.

Unifying practical uncertainty representations. II: Clouds

International Journal of Approximate …, 2008

There exist many simple tools for jointly capturing variability and incomplete information by means of uncertainty representations. Among them are random sets, possibility distributions, probability intervals, and the more recent Ferson's p-boxes and Neumaier's clouds, both defined by pairs of possibility distributions. In the companion paper, we have extensively studied a generalized form of p-box and situated it with respect to other models . This paper focuses on the links between clouds and other representations. Generalized p-boxes are shown to be clouds with comonotonic distributions. In general, clouds cannot always be represented by random sets, in fact not even by 2-monotone (convex) capacities.

Pari-mutuel probabilities as an uncertainty model

Information Sciences

The pari-mutuel model is a betting scheme that has its origins in horse racing, and that has been applied in a number of contexts, mostly economics. In this paper, we consider the set of probability measures compatible with a pari-mutuel model, characterize its extreme points, and investigate the properties of the associated lower and upper probabilities. We show that the pari-mutuel model can be embedded within the theory of probability intervals, and prove necessary and sucient conditions for it to be a belief function or a minitive measure. In addition, we also investigate the combination of dierent pari-mutuel models and their denition on product spaces.