Romano Scozzafava

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.

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.

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.

Conditional Events with Vague Information in Expert Systems

Ad hoc techniques and inference methods used in expert systems are often logically inconsistent. On the other hand, among properties and assertions concerning handling of uncertainty, those which turns out to be well founded can be in general easily deduced from probability laws. Relying on the general concept of event as a proposition and starting from a few conditional events of initial interest, a gradual and coherent assignment of conditional probabilities is possible by resorting to de Finettis theory of coherent extension of subjective probability. Moreover, even when numerical probabilities can be easily assessed, a more general approach is obtained introducing an ordering among conditional events by means of a coherent qualitative probability.

Inferential processes leading to possibility and necessity

This paper deals with the upper and lower bounds of a class of uncertainty measures endowed with particular characteristics (decomposability, monotonicity, partial additivity and so on). We consider an initial partial assessment consistent with either probability or possibility or necessity, then we study the upper and lower envelopes of all possible extensions. By resorting to a notion of weak logical independence we get as lower or upper envelope a possibility or a necessity, respectively, starting either from a probability or from a possibility or from a necessity.

Comparative probability for conditional events: A new look through coherence

We study, from the standpoint of coherence, comparative probabilities on an arbitrary familyE of conditional events. Given a binary relation ⩽·, coherence conditions on ⩽· are related to de Finettis coherent betting system: we consider their connections to the usual properties of comparative probability and to the possibility of numerical representations of ⩽·. In this context, the numerical reference frame is that of de Finettis coherent subjective conditional probability, which is not introduced (as in Kolmogoroffs approach) through a ratio between probability measures.Another relevant feature of our approach is that the family & need not have any particular algebraic structure, so that the ordering can be initially given for a few conditional events of interest and then possibly extended by a step-by-step procedure, preserving coherence.