Giuseppe Sanfilippo

Probabilistic logic under coherence: complexity and algorithms

In previous work [V. Biazzo, A. Gilio, T. Lukasiewicz and G. Sanfilippo, Probabilistic logic under coherence, model-theoretic probabilistic logic, and default reasoning in System P, Journal of Applied Non-Classical Logics 12(2) (2002) 189–213.], we have explored the relationship between probabilistic reasoning under coherence and model-theoretic probabilistic reasoning. In particular, we have shown that the notions of g-coherence and of g-coherent entailment in probabilistic reasoning under coherence can be expressed by combining notions in model-theoretic probabilistic reasoning with concepts from default reasoning. In this paper, we continue this line of research. Based on the above semantic results, we draw a precise picture of the computational complexity of probabilistic reasoning under coherence. Moreover, we introduce transformations for probabilistic reasoning under coherence, which reduce an instance of deciding g-coherence or of computing tight intervals under g-coherent entailment to a smaller problem instance, and which can be done very efficiently. Furthermore, we present new algorithms for deciding g-coherence and for computing tight intervals under g-coherent entailment, which reformulate previous algorithms using terminology from default reasoning. They are based on reductions to standard problems in model-theoretic probabilistic reasoning, which in turn can be reduced to linear optimization problems. Hence, efficient techniques for model-theoretic probabilistic reasoning can immediately be applied for probabilistic reasoning under coherence (for example, column generation techniques). We describe several such techniques, which transform problem instances in model-theoretic probabilistic reasoning into smaller problem instances. We also describe a technique for obtaining a reduced set of variables for the associated linear optimization problems in the conjunctive case, and give new characterizations of this reduced set as a set of non-decomposable variables, and using the concept of random gain.

Probabilistic logic under coherence, model-theoretic probabilistic logic, and default reasoning in System P

We study probabilistic logic under the viewpoint of the coherence principle of de Finetti. In detail, we explore how probabilistic reasoning under coherence is related to model-theoretic probabilistic reasoning and to default reasoning in System P. In particular, we show that the notions of g-coherence and of g-coherent entailment can be expressed by combining notions in model-theoretic probabilistic logic with concepts from default reasoning. Moreover, we show that probabilistic reasoning under coherence is a generalization of default reasoning in System P. That is, we provide a new probabilistic semantics for System P, which neither uses infinitesimal probabilities nor atomic bound (or big-stepped) probabilities. These results also provide new algorithms for probabilistic reasoning under coherence and for default reasoning in System P, and they give new insight into default reasoning with conditional objects.

Transitivity in coherence-based probability logic

We study probabilistically informative (weak) versions of transitivity by using suitable definitions of defaults and negated defaults in the setting of coherence and imprecise probabilities. We represent p-consistent sequences of defaults and/or negated defaults by g-coherent imprecise probability assessments on the respective sequences of conditional events. Moreover, we prove the coherent probability propagation rules for Weak Transitivity and the validity of selected inference patterns by proving p-entailment of the associated knowledge bases. Finally, we apply our results to study selected probabilistic versions of classical categorical syllogisms and construct a new version of the square of opposition in terms of defaults and negated defaults.

Probabilistic entailment in the setting of coherence: The role of quasi conjunction and inclusion relation

In this paper, by adopting a coherence-based probabilistic approach to default reasoning, we focus the study on the logical operation of quasi conjunction and the Goodman-Nguyen inclusion relation for conditional events. We recall that quasi conjunction is a basic notion for defining consistency of conditional knowledge bases. By deepening some results given in a previous paper we show that, given any finite family of conditional events F and any nonempty subset S of F, the family F p-entails the quasi conjunction C(S); then, given any conditional event E|H, we analyze the equivalence between p-entailment of E|H from F and p-entailment of E|H from C(S), where S is some nonempty subset of F We also illustrate some alternative theorems related with p-consistency and p-entailment. Finally, we deepen the study of the connections between the notions of p-entailment and inclusion relation by introducing for a pair (F,E|H) the (possibly empty) class K of the subsets S of F such that CS implies E|H. We show that the class K satisfies many properties; in particular K is additive and has a greatest element which can be determined by applying a suitable algorithm.

Probabilistic Logic under Coherence, Model-Theoretic Probabilistic Logic, and Default Reasoning

We study probabilistic logic under the viewpoint of the coherence principle of de Finetti. In detail, we explore the relationship between coherence-based and model-theoretic probabilistic logic. Interestingly, we show that the notions of g-coherence and of g-coherent entailment can be expressed by combining notions in model-theoretic probabilistic logic with concepts from default reasoning. Crucially, we even show that probabilistic reasoning under coherence is a probabilistic generalization of default reasoning in system P. That is, we provide a new probabilistic semantics for system P, which is neither based on infinitesimal probabilities nor on atomic-bound (or also big-stepped) probabilities. These results also give new insight into default reasoning with conditional objects.

Coherence checking and propagation of lower probability bounds

Abstract In this paper we use imprecise probabilities, based on a concept of generalized coherence (g-coherence), for the management of uncertain knowledge and vague information. We face the problem of reducing the computational difficulties in g-coherence checking and propagation of lower conditional probability bounds. We examine a procedure, based on linear systems with a reduced number of unknowns, for the checking of g-coherence. We propose an iterative algorithm to determine the reduced linear systems. Based on the same ideas, we give an algorithm for the propagation of lower probability bounds. We also give some theoretical results that allow, by suitably modifying our algorithms, the g-coherence checking and propagation by working with a reduced set of variables and/or with a reduced set of constraints. Finally, we apply our algorithms to some examples.

Extropy: Complementary Dual of Entropy

This article provides a completion to theories of information based on entropy, resolving a longstanding question in its axiomatization as proposed by Shannon and pursued by Jaynes. We show that Shannon’s entropy function has a complementary dual function which we call “extropy.” The entropy and the extropy of a binary distribution are identical. However, the measure bifurcates into a pair of distinct measures for any quantity that is not merely an event indicator. As with entropy, the maximum extropy distribution is also the uniform distribution, and both measures are invariant with respect to permutations of their mass functions. However, they behave quite differently in their assessments of the refinement of a distribution, the axiom which concerned Shannon and Jaynes. Their duality is specified via the relationship among the entropies and extropies of course and fine partitions. We also analyze the extropy function for densities, showing that relative extropy constitutes a dual to the Kullback–Leibler divergence, widely recognized as the continuous entropy measure. These results are unified within the general structure of Bregman divergences. In this context they identify half the L2 metric as the extropic dual to the entropic directed distance. We describe a statistical application to the scoring of sequential forecast distributions which provoked the discovery.

Conjunction, Disjunction and Iterated Conditioning of Conditional Events

Starting from a recent paper by S. Kaufmann, we introduce a notion of conjunction of two conditional events and then we analyze it in the setting of coherence. We give a representation of the conjoined conditional and we show that this new object is a conditional random quantity, whose set of possible values normally contains the probabilities assessed for the two conditional events. We examine some cases of logical dependencies, where the conjunction is a conditional event; moreover, we give the lower and upper bounds on the conjunction. We also examine an apparent paradox concerning stochastic independence which can actually be explained in terms of uncorrelation. We briefly introduce the notions of disjunction and iterated conditioning and we show that the usual probabilistic properties still hold.

Coherent Conditional Previsions and Proper Scoring Rules

In this paper we study the relationship between the notion of coherence for conditional prevision assessments on a family of finite conditional random quantities and the notion of admissibility with respect to bounded strictly proper scoring rules. Our work extends recent results given by the last two authors of this paper on the equivalence between coherence and admissibility for conditional probability assessments. In order to prove that admissibility implies coherence a key role is played by the notion of Bregman divergence.

Centering and Compound Conditionals Under Coherence

There is wide support in logic , philosophy , and psychology for the hypothesis that the probability of the indicative conditional of natural language, \(P(\textit{if } A \textit{ then } B)\), is the conditional probability of B given A, P(B|A). We identify a conditional which is such that \(P(\textit{if } A \textit{ then } B)= P(B|A)\) with de Finetti’s conditional event, B|A. An objection to making this identification in the past was that it appeared unclear how to form compounds and iterations of conditional events. In this paper, we illustrate how to overcome this objection with a probabilistic analysis, based on coherence, of these compounds and iterations. We interpret the compounds and iterations as conditional random quantities, which sometimes reduce to conditional events, given logical dependencies. We also show, for the first time, how to extend the inference of centering for conditional events, inferring B|A from the conjunction A and B, to compounds and iterations of both conditional events and biconditional events, B||A, and generalize it to n-conditional events.