702 research outputs found
Subjective probability, trivalent logics and compound conditionals
In this work we first illustrate the subjective theory of de Finetti. We
recall the notion of coherence for both the betting scheme and the penalty
criterion, by considering the unconditional and conditional cases. We show the
equivalence of the two criteria by giving the geometrical interpretation of
coherence. We also consider the notion of coherence based on proper scoring
rules. We discuss conditional events in the trivalent logic of de Finetti and
the numerical representation of truth-values. We check the validity of selected
basic logical and probabilistic properties for some trivalent logics:
Kleene-Lukasiewicz-Heyting-de Finetti; Lukasiewicz; Bochvar-Kleene; Sobocinski.
We verify that none of these logics satisfies all the properties. Then, we
consider our approach to conjunction and disjunction of conditional events in
the setting of conditional random quantities. We verify that all the basic
logical and probabilistic properties (included the Fr\'{e}chet-Hoeffding
bounds) are preserved in our approach. We also recall the characterization of
p-consistency and p-entailment by our notion of conjunction
The Goodman-Nguyen Relation within Imprecise Probability Theory
The Goodman-Nguyen relation is a partial order generalising the implication
(inclusion) relation to conditional events. As such, with precise probabilities
it both induces an agreeing probability ordering and is a key tool in a certain
common extension problem. Most previous work involving this relation is
concerned with either conditional event algebras or precise probabilities. We
investigate here its role within imprecise probability theory, first in the
framework of conditional events and then proposing a generalisation of the
Goodman-Nguyen relation to conditional gambles. It turns out that this relation
induces an agreeing ordering on coherent or C-convex conditional imprecise
previsions. In a standard inferential problem with conditional events, it lets
us determine the natural extension, as well as an upper extension. With
conditional gambles, it is useful in deriving a number of inferential
inequalities.Comment: Published version:
http://www.sciencedirect.com/science/article/pii/S0888613X1400101
Default Logic in a Coherent Setting
In this talk - based on the results of a forthcoming paper (Coletti,
Scozzafava and Vantaggi 2002), presented also by one of us at the Conference on
"Non Classical Logic, Approximate Reasoning and Soft-Computing" (Anacapri,
Italy, 2001) - we discuss the problem of representing default rules by means of
a suitable coherent conditional probability, defined on a family of conditional
events. An event is singled-out (in our approach) by a proposition, that is a
statement that can be either true or false; a conditional event is consequently
defined by means of two propositions and is a 3-valued entity, the third value
being (in this context) a conditional probability
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 C(S)
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
Coherent Risk Measures and Upper Previsions
In this paper coherent risk measures and other currently used risk measures, notably Value-at-Risk (VaR), are studied from the perspective of the theory of coherent imprecise previsions. We introduce the notion of coherent risk measure defined on an arbitrary set of risks, showing that it can be considered a special case of coherent upper prevision. We also prove that our definition generalizes the notion of coherence for risk measures defined on a linear space of random numbers, given in literature. We also show that Value-at-Risk does not necessarily satisfy a weaker notion of coherence called âavoiding sure lossâ (ASL), and discuss both sufficient conditions for VaR to avoid sure loss and ways of modifying VaR into a coherent risk measure.Coherent risk measure, imprecise prevision, Value-at-Risk, avoiding sure loss condition
Nonmonotonic Probabilistic Logics between Model-Theoretic Probabilistic Logic and Probabilistic Logic under Coherence
Recently, it has been shown that probabilistic entailment under coherence is
weaker than model-theoretic 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 model-theoretic probabilistic entailment and probabilistic
entailment under coherence. That is, the new formalisms properly generalize
their counterparts in classical default reasoning, they are weaker than
model-theoretic 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
well-known standard ones.Comment: 10 pages; in Proceedings of the 9th International Workshop on
Non-Monotonic Reasoning (NMR-2002), Special Session on Uncertainty Frameworks
in Nonmonotonic Reasoning, pages 265-274, Toulouse, France, April 200
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