Giulianella Coletti

Probabilistic logic in a coherent setting

1. Introduction. 2. Events as Propositions. 3. Finitely Additive Probability. 4. Coherent probability. 5. Betting Interpretation of Coherence. 6. Coherent Extensions of Probability Assessments. 7. Random Quantities. 8. Probability Meaning and Assessment: a Reconciliation. 9. To Be or not To Be Compositional? 10. Conditional Events. 11. Coherent Conditional Probability. 12. Zero-Layers. 13. Coherent Extensions of Conditional Probability. 14. Exploiting Zero Probabilities. 15. Lower and Upper Conditional Probabilities. 16. Inference. 17. Stochastic Independence in a Coherent Setting. 18. A Random Walk in the Midst of Paradigmatic Examples. 19. Fuzzy Sets and Possibility as Coherent Conditional Probabilities. 20. Coherent Conditional Probability and Default Reasoning. 21. A Short Account of Decomposable Measures of Uncertainty. Bibliography. Index.

Coherent numerical and ordinal probabilistic assessments

Conditions of coherence are given for generalized assessments of probability on arbitrary sets of conditional events, that is assessments including also imprecise values (probability intervals) and ordinal evaluations (comparative probabilities). Such coherence conditions ensure, like the well known de Finetti coherence condition for numerical probabilities, the possibility of extending generalized assessments of probability and preserving coherence. The main results demonstrate that the introduced coherence conditions are necessary and sufficient for the existence of a de Finetti coherent probability, agreeing with the generalized probabilistic assessment. >

CHARACTERIZATION OF COHERENT CONDITIONAL PROBABILITIES AS A TOOL FOR THEIR ASSESSMENT AND EXTENSION

A major purpose of this paper is to show the broad import and applicability of the theory of probability as proposed by de Finetti, which differs radically from the usual one (based on a measure-theoretic framework). In particular, with reference to a coherent conditional probability, we prove a characterization theorem, which provides also a useful algorithm for checking coherence of a given assessment. Moreover it allows to deepen and generalise in useful directions de Finetti’s extension theorem (dubbed as “the fundamental theorem of probability”), emphasising its operational aspects in many significant applications.

Conditional probability, fuzzy sets, and possibility: a unifying view

We propose an interpretation of fuzzy set theory (both from a semantic and a syntactic point of view) in terms of conditional events and coherent conditional probabilities. During past years, a large number of papers has been devoted to support either the thesis that probability theory is all that is required for reasoning about uncertainty, or the negative view maintaining that probability is inadequate to capture what is usually treated by fuzzy theory. In this paper we emphasize the role of conditioning (in a proper framework, i.e. de Finettis coherence) to get rid of many controversial aspects. Moreover, we introduce suitable operations between fuzzy subsets, looked on as corresponding operations between conditional events endowed with the relevant conditional probability. Finally, we show how the concept of possibility function naturally arises as a coherent conditional probability.

Conditioning and inference in intelligent systems

Abstract We maintain that among the research trends concerning the various aspects and methodologies for the management of partial and revisable information in automated reasoning and giving particular emphasis to conditioning and inference, conditional events and conditional probability (in a coherent– in the sense of de Finetti – framework) play a central role: we will review some of our and related results, showing that this approach is able to encompass many of other existing numerical and symbolic methods dealing with the treatment of partial knowledge and uncertainty in Artificial Intelligence.

Independence and Possibilistic Conditioning

There is not a unique definition of a conditional possibility distribution since the concept of conditioning is complex and many papers have been conducted to define conditioning in a possibilistic framework. In most cases, independence has been also defined and studied by means of a kind of analogy with the probabilistic case. In [2,4], we introduce conditional possibility as a primitive concept by means of a function whose domain is a set of conditional events. In this paper, we define a concept of independence associated with this form of conditional possibility and we show that classical properties required for independence concepts are satisfied.

From Conditional Events to Conditional Measures: A New Axiomatic Approach

Our starting point is a definition of conditional event E❘H which differs from many seemingly “similar” ones adopted in the relevant literature since 1935, starting with de Finetti. In fact, if we do not assign the same “third” value u (“undetermined”) to all conditional events, but make it depend on E❘H, it turns out that this function t(E❘H) can be taken as a general conditional uncertainty measure, and we get (through a suitable – in a sense, “compulsory” – choice of the relevant operations among conditional events) the “natural” axioms for many different (besides probability) conditional measures.

Mathematical models for handling partial knowledge in artificial intelligence

Invited Papers: Ellsberg Paradox Intuition and Choquet Expected Utility (A. Chateauneuf). Fuzzy Logic as Logic (P. Hajek). Mathematical Foundations of Evidence Theory (J. Kohlas). Semantics for Uncertain Inference Based on Statistical Knowledge (H.E. Kyburg). Prospects and Problems in Applying the Fundamental Theorem of Prevision as an Expert System: An Example of Learning about Parole Decisions (F. Lad, I. Coope). Coherent Prevision as a Linear Functional without an Underlying Measure Space: The Purely Arithmetic Structure of Logical Relations among Conditional Quantities (F. Lad). Revision Rules for Convex Sets of Probabilities (S. Moral, N. Wilson). Contributed Papers: Generalized Concept of Atoms for Conditional Events (A. Capotorti). Checking the Coherence of Conditional Probabilities in Expert Systems: Remarks and Algorithms (G. Di Biase, A. Maturo). A Hyperstructure of Conditional Events for Artificial Intelligence (S. Doria, A. Maturo). Possibilistic Logic and Plausible Inference (D. Dubois, H. Prade). Probability Logic as a Fuzzy Logic (G. Gerla). Algorithms for Precise and Imprecise Conditional Probability Assessments (A. Gilio). A Valuationbased Architecture for Assumptionbased Reasoning (R. Haenni). 6 additional articles. Index.

Toward a general theory of conditional beliefs

We consider a class of general decomposable measures of uncertainty, which encompasses (as its most specific elements, with respect to the properties of the rules of composition) probabilities, and (as its most general elements) belief functions. The aim, using this general context, is to introduce (in a direct way) the concept of conditional belief function as a conditional generalized decomposable measure φ(·|·), defined on a set of conditional events. Our main tool will be the following result, that we prove in the first part of the article and which is a sort of converse of a well‐known result (i.e., a belief function is a lower probability): a coherent conditional lower probability P(·|K) extending a coherent probability P(Hi)—where the events His are a partition of the certain event Ω and K is the union of some (possibly all) of them—is a belief function.

Coherent qualitative probability

Abstract This note introduces the concepts of coherent, positive coherent, and strongly coherent qualitative probability. The first interesting result is that such a qualitative probability can be extended from a given domain (not necessarily an algebra) to an arbitrary larger one. The most important result of this paper consists in proving that coherent qualitative probabilities can be represented by the Finettis coherent previsions.