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25
Probabilistic Logic under Coherence: Complexity and Algorithms
 In Proceedings ISIPTA01
, 2001
"... We study probabilistic logic under the viewpoint of the coherence principle of de Finetti. In detail, we explore the relationship between coherencebased and classical modeltheoretic probabilistic logic. Interestingly, we show that the notions of gcoherence and of gcoherent entailment can be expre ..."
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Cited by 22 (11 self)
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We study probabilistic logic under the viewpoint of the coherence principle of de Finetti. In detail, we explore the relationship between coherencebased and classical modeltheoretic probabilistic logic. Interestingly, we show that the notions of gcoherence and of gcoherent entailment can be expressed by combining notions in modeltheoretic probabilistic logic with concepts from default reasoning. Using these results, we analyze the computational complexity of probabilistic reasoning under coherence. Moreover, we present new algorithms for deciding gcoherence and for computing tight gcoherent intervals, which reduce these tasks to standard reasoning tasks in modeltheoretic probabilistic logic. Thus, efficient techniques for modeltheoretic probabilistic reasoning can immediately be applied for probabilistic reasoning under coherence, for example, column generation techniques. We then describe two other interesting techniques for efficient modeltheoretic probabilistic reasoning in the conjunctive case.
Probabilistic Logic under Coherence, ModelTheoretic Probabilistic Logic, and Default Reasoning
 Journal of Applied NonClassical Logics
"... We study probabilistic logic under the viewpoint of the coherence principle of de Finetti. In detail, we explore the relationship between coherencebased and modeltheoretic probabilistic logic. Interestingly, we show that the notions of gcoherence and of gcoherent entailment can be expressed by co ..."
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Cited by 22 (9 self)
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We study probabilistic logic under the viewpoint of the coherence principle of de Finetti. In detail, we explore the relationship between coherencebased and modeltheoretic probabilistic logic. Interestingly, we show that the notions of gcoherence and of gcoherent entailment can be expressed by combining notions in modeltheoretic 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 atomicbound (or also bigstepped) probabilities. These results also give new insight into default reasoning with conditional objects.
Weak nonmonotonic probabilistic logics
"... Towards probabilistic formalisms for resolving local inconsistencies under modeltheoretic probabilistic entailment, we present probabilistic generalizations of Pearl’s entailment in System Z and Lehmann’s lexicographic entailment. We then analyze the nonmonotonic and semantic properties of the new ..."
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Cited by 21 (6 self)
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Towards probabilistic formalisms for resolving local inconsistencies under modeltheoretic probabilistic entailment, we present probabilistic generalizations of Pearl’s entailment in System Z and Lehmann’s lexicographic entailment. We then analyze the nonmonotonic and semantic properties of the new notions of entailment. In particular, we show that they satisfy the rationality postulates of System P and the property of Rational Monotonicity. Moreover, we show that modeltheoretic probabilistic entailment is stronger than the new notion of lexicographic entailment, which in turn is stronger than the new notion of entailment in System Z. As an important feature of the new notions of entailment in System Z and lexicographic entailment, we show that they coincide with modeltheoretic probabilistic entailment whenever there are no local inconsistencies. We also show that the new notions of entailment in System Z and lexicographic entailment are proper generalizations of their classical counterparts. Finally, we present algorithms for reasoning under the new formalisms, and we give a precise picture of its computational complexity.
Aggregating disparate estimates of chance
, 2004
"... We consider a panel of experts asked to assign probabilities to events, both logically simple and complex. The events evaluated by different experts are based on overlapping sets of variables but may otherwise be distinct. The union of all the judgments will likely be probabilistic incoherent. We ad ..."
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Cited by 19 (4 self)
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We consider a panel of experts asked to assign probabilities to events, both logically simple and complex. The events evaluated by different experts are based on overlapping sets of variables but may otherwise be distinct. The union of all the judgments will likely be probabilistic incoherent. We address the problem of revising the probability estimates of the panel so as to produce a coherent set that best represents the group’s expertise.
Inference in conditional probability logic
 Kybernetika
, 2006
"... An important field of probability logic is the investigation of inference rules that propagate point probabilities or, more generally, interval probabilities from premises to conclusions. Conditional probability logic (CPL) interprets the common sense expressions of the form “if..., then... ” by con ..."
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Cited by 13 (10 self)
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An important field of probability logic is the investigation of inference rules that propagate point probabilities or, more generally, interval probabilities from premises to conclusions. Conditional probability logic (CPL) interprets the common sense expressions of the form “if..., then... ” by conditional probabilities and not by the probability of the material implication. An inference rule is probabilistically informative if the coherent probability interval of its conclusion is not necessarily equal to the unit interval [0, 1]. Not all logically valid inference rules are probabilistically informative and vice versa. The relationship between logically valid and probabilistically informative inference rules is discussed and illustrated by examples such as the modus ponens or the affirming the consequent. We propose a method to evaluate the strength of CPL inference rules. Finally, an example of a proof is given that is purely based on CPL inference rules.
Possible worlds semantics for probabilistic logic programs
 ICLP 2004
, 2004
"... Abstract. In this paper we consider a logic programming framework for reasoning about imprecise probabilities. In particular, we propose a new semantics, for the Probabilistic Logic Programs (pprograms) of Ng and Subrahmanian. Pprograms represent imprecision using probability intervals. Our seman ..."
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Cited by 12 (2 self)
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Abstract. In this paper we consider a logic programming framework for reasoning about imprecise probabilities. In particular, we propose a new semantics, for the Probabilistic Logic Programs (pprograms) of Ng and Subrahmanian. Pprograms represent imprecision using probability intervals. Our semantics, based on the possible worlds semantics, considers all point probability distributions that satisfy a given pprogram. In the paper, we provide the exact characterization of such models of a pprogram. We show that the set of models of a pprogram cannot, in general case, be described by single intervals associated with atoms of the program. We provide algorithms for efficient construction of this set of models and study their complexity. 1
Nonmonotonic Probabilistic Logics between ModelTheoretic Probabilistic Logic and Probabilistic Logic under Coherence
, 2002
"... Recently, it has been shown that probabilistic entailment under coherence is weaker than modeltheoretic probabilistic entailment. Moreover, probabilistic entailment under coherence is a generalization of default entailment in System P. In this paper, we continue this line of research by presenting ..."
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Cited by 7 (6 self)
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Recently, it has been shown that probabilistic entailment under coherence is weaker than modeltheoretic probabilistic entailment. Moreover, probabilistic entailment under coherence is a generalization of default entailment in System P. In this paper, we continue this line of research by presenting probabilistic generalizations of more sophisticated notions of classical default entailment that lie between modeltheoretic probabilistic entailment and probabilistic entailment under coherence. That is, the new formalisms properly generalize their counterparts in classical default reasoning, they are weaker than modeltheoretic probabilistic entailment, and they are stronger than probabilistic entailment under coherence. The new formalisms are useful especially for handling probabilistic inconsistencies related to conditioning on zero events. They can also be applied for probabilistic belief revision. More generally, in the same spirit as a similar previous paper, this paper sheds light on exciting new formalisms for probabilistic reasoning beyond the wellknown standard ones.
Databases for Interval Probabilities
, 2004
"... We present a database framework for the efficient storage and manipulation of interval probability distributions and their associated information. While work on interval probabilities and on probabilistic databases has appeared before, ours is the first to combine these into a coherent and mathemati ..."
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Cited by 4 (0 self)
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We present a database framework for the efficient storage and manipulation of interval probability distributions and their associated information. While work on interval probabilities and on probabilistic databases has appeared before, ours is the first to combine these into a coherent and mathematically sound framework including both standard relational queries and queries based on probability theory. In particular, our query algebra allows users not only to query existing interval probability distributions, but also to construct new ones by means of conditionalization and marginalization, as well as other more common database
Nonmonotonic probabilistic reasoning under variablestrength inheritance with overriding
 SYNTHESE
, 2005
"... We present new probabilistic generalizations of Pearl’s entailment in System ..."
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Cited by 3 (2 self)
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We present new probabilistic generalizations of Pearl’s entailment in System
Convenient Interactive Computing for Coherent Imprecise Prevision Assessments
 Carleton Scientific
, 2003
"... A generalization of deFinetti's Fundamental Theorem of Probability facilitates coherent assessment, by iterated natural extension, of imprecise probabilities or expectations, conditional and unconditional. Point values are generalized to assessed bounds, accepted under weak coherence, that is, al ..."
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Cited by 3 (0 self)
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A generalization of deFinetti's Fundamental Theorem of Probability facilitates coherent assessment, by iterated natural extension, of imprecise probabilities or expectations, conditional and unconditional. Point values are generalized to assessed bounds, accepted under weak coherence, that is, allowing the input of redundant loose bounds. The method is realized in a convenient interactive computer program, which is demonstrated here, and made available as open source code. This work suggests that a consulting expert's fees should not be paid unless his/her assessed probabilities cohere.