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Logics of Imprecise Comparative Probability
This paper studies connections between two alternatives to the standard probability calculus for representing and reasoning about uncertainty: imprecise probability andcomparative probability. The goal is to identify complete logics for reasoning about uncertainty in a comparative probabilistic language whose semantics is given in terms of imprecise probability. Comparative probability operators are interpreted as quantifying over a set of probability measures. Modal and dynamic operators are added for reasoning about epistemic possibility and updating sets of probability measures
Lexicographic choice functions
We investigate a generalisation of the coherent choice functions considered
by Seidenfeld et al. (2010), by sticking to the convexity axiom but imposing no
Archimedeanity condition. We define our choice functions on vector spaces of
options, which allows us to incorporate as special cases both Seidenfeld et
al.'s (2010) choice functions on horse lotteries and sets of desirable gambles
(Quaeghebeur, 2014), and to investigate their connections. We show that choice
functions based on sets of desirable options (gambles) satisfy Seidenfeld's
convexity axiom only for very particular types of sets of desirable options,
which are in a one-to-one relationship with the lexicographic probabilities. We
call them lexicographic choice functions. Finally, we prove that these choice
functions can be used to determine the most conservative convex choice function
associated with a given binary relation.Comment: 27 page
Decision-Making with Belief Functions: a Review
Approaches to decision-making under uncertainty in the belief function
framework are reviewed. Most methods are shown to blend criteria for decision
under ignorance with the maximum expected utility principle of Bayesian
decision theory. A distinction is made between methods that construct a
complete preference relation among acts, and those that allow incomparability
of some acts due to lack of information. Methods developed in the imprecise
probability framework are applicable in the Dempster-Shafer context and are
also reviewed. Shafer's constructive decision theory, which substitutes the
notion of goal for that of utility, is described and contrasted with other
approaches. The paper ends by pointing out the need to carry out deeper
investigation of fundamental issues related to decision-making with belief
functions and to assess the descriptive, normative and prescriptive values of
the different approaches
Accept & Reject Statement-Based Uncertainty Models
We develop a framework for modelling and reasoning with uncertainty based on
accept and reject statements about gambles. It generalises the frameworks found
in the literature based on statements of acceptability, desirability, or
favourability and clarifies their relative position. Next to the
statement-based formulation, we also provide a translation in terms of
preference relations, discuss---as a bridge to existing frameworks---a number
of simplified variants, and show the relationship with prevision-based
uncertainty models. We furthermore provide an application to modelling symmetry
judgements.Comment: 35 pages, 17 figure
Reasons and Means to Model Preferences as Incomplete
Literature involving preferences of artificial agents or human beings often
assume their preferences can be represented using a complete transitive binary
relation. Much has been written however on different models of preferences. We
review some of the reasons that have been put forward to justify more complex
modeling, and review some of the techniques that have been proposed to obtain
models of such preferences
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
The shape of incomplete preferences
Incomplete preferences provide the epistemic foundation for models of
imprecise subjective probabilities and utilities that are used in robust
Bayesian analysis and in theories of bounded rationality. This paper presents a
simple axiomatization of incomplete preferences and characterizes the shape of
their representing sets of probabilities and utilities. Deletion of the
completeness assumption from the axiom system of Anscombe and Aumann yields
preferences represented by a convex set of state-dependent expected utilities,
of which at least one must be a probability/utility pair. A strengthening of
the state-independence axiom is needed to obtain a representation purely in
terms of a set of probability/utility pairs.Comment: Published at http://dx.doi.org/10.1214/009053606000000740 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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