Veronica Biazzo

A generalization of the fundamental theorem of de Finetti for imprecise conditional probability assessments

Abstract In this paper, we consider coherent imprecise probability assessments on finite families of conditional events and we study the problem of their extension. With this aim, we adopt a generalized definition of coherence, called g -coherence, which is based on a suitable generalization of the coherence principle of de Finetti. At first, we recall some theoretical results and an algorithm obtained in some previous papers where the case of precise conditional probability assessments has been studied. Then, we extend these results to the case of imprecise probabilistic assessments and we obtain a theorem which can be looked at as a generalization of the version of the fundamental theorem of de Finetti given by some authors for the case of conditional events. Our algorithm can also be exploited to produce lower and upper probabilities which are coherent in the sense of Walley and Williams. Moreover, we compare our approach to similar ones, like probability logic or probabilistic deduction. Finally, we apply our algorithm to some well-known inference rules assuming some logical relations among the given events.

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.

On the Linear Structure of Betting Criterion and the Checking of Coherence

We use imprecise probabilities, based on a concept of generalized coherence, for the management of uncertainty in artificial intelligence. With the aim of reducing the computational difficulties, in the checking of generalized coherence we propose a method which exploits, in the framework of the betting criterion, suitable subsets of the sets of values of the random gains. We give an algorithm in each step of which a linear system with a reduced number of unknowns can be used. Our method improves a procedure already existing in literature and could be integrated with recent approaches of other authors, who exploit suitable logical conditions with the aim of splitting the problem into subproblems. We remark that our approach could be also used in combination with efficient methods like column generation techniques. Finally, to illustrate our method, we give some examples.

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.

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.

Some theoretical properties of conditional probability assessments

We consider a finite family of conditional events and, among other results, we prove a connection property for the set of coherent assessments on such family. This property assures that, for every pair of coherent assessments on the family, there exists (at least) a continuous curve C whose points are intermediate coherent probability assessments. We also consider the compactness property for the set of coherent assessments. Then, as a corollary of connection and closure properties, we obtain the theorem of extension for coherent conditional probabilities.

Temporal probabilistic object bases

There are numerous applications where we have to deal with temporal uncertainty associated with objects. The ability to automatically store and manipulate time, probabilities, and objects is important. We propose a data model and algebra for temporal probabilistic object bases (TPOBs), which allows us to specify the probability with which an event occurs at a given time point. In explicit TPOB-instances, the sets of time points along with their probability intervals are explicitly enumerated. In implicit TPOB-instances, sets of time points are expressed by constraints and their probability intervals by probability distribution functions. Thus, implicit object base instances are succinct representations of explicit ones; they allow for an efficient implementation of algebraic operations, while their explicit counterparts make defining algebraic operations easy. We extend the relational algebra to both explicit and implicit instances and prove that the operations on implicit instances correctly implement their counterpart on explicit instances.