249 research outputs found
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
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
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.Comment: Published at http://dx.doi.org/10.1214/14-STS430 in the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
La Ruota della Fortuna
Nell’anno scolastico 2013-2014 presso l’I. C. Boccadifalco Tomasi di Lampedusa di Palermo abbiamo realizzato un progetto PON finalizzato all’ampliamento della matematica per le classi seconde della scuola primaria. In una delle attività realizzate, che descriviamo in questo lavoro, ci siamo occupati del problema dell’elicitazione delle probabilità basata sul criterio della scommessa [1]. Nell’attività proposta si richiede la formulazione delle probabilità mediante dei gradi di fiducia su alcuni eventi relativi al gioco della ruota della fortuna. In particolare, si propone allo studente di formulare delle valutazioni di probabilità mediante delle scommesse nel ruolo di giocatore in un gioco inizialmente non equo. Dopo aver fatto un consistente numero di giri della ruota, si commentano eventuali risultati a favore del banco e si osserva che per rendere il gioco equo occorre modificare le quote di vincita in modo tale che esse coincidano con i reciproci delle probabilità. Per l’azione didattica proposta si può far ricorso a modalità di apprendimento cooperativo. Per non far apparire arida, astrusa ed inutile la matematica, si può far vedere che essa è lo strumento naturale per affrontare, analizzare e risolvere, problemi anche in situazioni di incertezza o conoscenza parziale
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
Quasi Conjunction, Quasi Disjunction, T-norms and T-conorms: Probabilistic Aspects
We make a probabilistic analysis related to some inference rules which play
an important role in nonmonotonic reasoning. In a coherence-based setting, we
study the extensions of a probability assessment defined on conditional
events to their quasi conjunction, and by exploiting duality, to their quasi
disjunction. The lower and upper bounds coincide with some well known t-norms
and t-conorms: minimum, product, Lukasiewicz, and Hamacher t-norms and their
dual t-conorms. On this basis we obtain Quasi And and Quasi Or rules. These are
rules for which any finite family of conditional events p-entails the
associated quasi conjunction and quasi disjunction. We examine some cases of
logical dependencies, and we study the relations among coherence, inclusion for
conditional events, and p-entailment. We also consider the Or rule, where quasi
conjunction and quasi disjunction of premises coincide with the conclusion. We
analyze further aspects of quasi conjunction and quasi disjunction, by
computing probabilistic bounds on premises from bounds on conclusions. Finally,
we consider biconditional events, and we introduce the notion of an
-conditional event. Then we give a probabilistic interpretation for a
generalized Loop rule. In an appendix we provide explicit expressions for the
Hamacher t-norm and t-conorm in the unitary hypercube
INTELLI 2013, The Second International Conference on Intelligent Systems and Applications
The research of optimization techniques in the system of goods distribution from warehouses to final users (vehicle routing problem), made considerable savings on the total cost of transport and, consequently, on the final cost of goods, and produced the models applicable to other operating environments (e.g., transport for disabled people, school, municipal waste collection). The analysis conducted on the different models developed under the VRP highlights the support that these models can give on the infomobility of goods
Connexive Logic, Probabilistic Default Reasoning, and Compound Conditionals
We present two approaches to investigate the validity of connexive principles and related formulas and properties within coherence-based probability logic. Connexive logic emerged from the intuition that conditionals of the form if not-A, then A, should not hold, since the conditional’s antecedent not-A contradicts its consequent A. Our approaches cover this intuition by observing that the only coherent probability assessment on the conditional event A | not-A is p(A | not-A) = 0. In the first approach we investigate connexive principles within coherence-based probabilistic default reasoning, by interpreting defaults and negated defaults in terms of suitable probabilistic constraints on conditional events. In the second approach we study connexivity within the coherence framework of compound conditionals, by interpreting connexive principles in terms of suitable conditional random quantities. After developing notions of validity in each approach, we analyze the following connexive principles: Aristotle’s theses, Aristotle’s Second Thesis, Abelard’s First Principle, and Boethius’ theses. We also deepen and generalize some principles and investigate further properties related to connexive logic (like non-symmetry). Both approaches satisfy minimal requirements for a connexive logic. Finally, we compare both approaches conceptually
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