32 research outputs found
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
From imprecise probability assessments to conditional probabilities with quasi additive classes of conditioning events
In this paper, starting from a generalized coherent (i.e. avoiding uniform
loss) intervalvalued probability assessment on a finite family of conditional
events, we construct conditional probabilities with quasi additive classes of
conditioning events which are consistent with the given initial assessment.
Quasi additivity assures coherence for the obtained conditional probabilities.
In order to reach our goal we define a finite sequence of conditional
probabilities by exploiting some theoretical results on g-coherence. In
particular, we use solutions of a finite sequence of linear systems.Comment: Appears in Proceedings of the Twenty-Eighth Conference on Uncertainty
in Artificial Intelligence (UAI2012
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
Deductive Reasoning Under Uncertainty Using a Water Tank Analogy
This paper describes a cubic water tank equipped with a movable partition receiving various amounts of liquid used to represent joint probability distributions. This device is applied to the investigation of deductive inferences under uncertainty. The analogy is exploited to determine by qualitative reasoning the limits in probability of the conclusion of twenty basic deductive arguments (such as Modus Ponens, And-introduction, Contraposition, etc.) often used as benchmark problems by the various theoretical approaches to reasoning under uncertainty. The probability bounds imposed by the premises on the conclusion are derived on the basis of a few trivial principles such as "a part of the tank cannot contain more liquid than its capacity allows", or "if a part is empty, the other part contains all the liquid". This stems from the equivalence between the physical constraints imposed by the capacity of the tank and its subdivisions on the volumes of liquid, and the axioms and rules of probability. The device materializes de Finetti's coherence approach to probability. It also suggests a physical counterpart of Dutch book arguments to assess individuals' rationality in probability judgments in the sense that individuals whose degrees of belief in a conclusion are out of the bounds would commit themselves to executing physically impossible tasks
Non-parametric probability distributions embedded inside of a linear space provided with a quadratic metric
There exist uncertain situations in which a random event is not a measurable set, but it is a point of a
linear space inside of which it is possible to study different random quantities characterized by non-parametric
probability distributions. We show that if an event is not a measurable set then it is contained in a closed
structure which is not a σ-algebra but it is a linear space over R. We think of probability as being a mass. It is
really a mass with respect to problems of statistical sampling. It is a mass with respect to problems of social
sciences. In particular, it is a mass with regard to economic situations studied by means of the subjective notion
of utility. We are able to decompose a random quantity meant as a geometric entity inside of a metric space.
It is also possible to decompose its prevision and variance inside of it. We show a quadratic metric in order
to obtain the variance of a random quantity. The origin of the notion of variability is not standardized within
this context. It always depends on the state of information and knowledge of an individual. We study different
intrinsic properties of non-parametric probability distributions as well as of probabilistic indices summarizing
them. We define the notion of α-distance between two non-parametric probability distributio
Probabilistic entailment and iterated conditionals
In this paper we exploit the notions of conjoined and iterated conditionals,
which are defined in the setting of coherence by means of suitable conditional
random quantities with values in the interval . We examine the iterated
conditional , by showing that p-entails if and only if
. Then, we show that a p-consistent family
p-entails a conditional event if
and only if , or for some nonempty
subset of , where is the quasi
conjunction of the conditional events in . Then, we examine the
inference rules , , , and of System~P
and other well known inference rules ( , ,
). We also show that , where
is the conjunction of the conditional events in
. We characterize p-entailment by showing that
p-entails if and only if .
Finally, we examine \emph{Denial of the antecedent} and \emph{Affirmation of
the consequent}, where the p-entailment of from does
not hold, by showing that $(E_3|H_3)|\mathcal{C}(\mathcal{F})\neq1.
On compound and iterated conditionals
We illustrate the notions of compound and iterated conditionals introduced, in recent papers, as suitable conditional random quantities, in the framework of coherence. We motivate our definitions by examining some concrete examples. Our logical operations among conditional events satisfy the basic probabilistic properties valid for unconditional events. We show that some, intuitively acceptable, compound sentences on conditionals can be analyzed in a rigorous way in terms of suitable iterated conditionals. We discuss the Import-Export principle, which is not valid in our approach, by also examining the inference from a material conditional to the associated conditional event. Then, we illustrate the characterization, in terms of iterated conditionals, of some well known p-valid and non p-valid inference rules