This article introduces a new wa of understanding subjective probabilit and its generalization to lower and upper prevision.

Current dynamic-epistemic logics model different types of information change in multi-agent scenarios. We propose a way to generalize these logics to a probabilistic setting, obtaining a calculus for multi-agent update with different slots for probability, and a matching dynamic logic of information change that has a probabilistic character itself. We present a general completeness result that not only holds for the particular logical system set out in this paper, but for a larger class of dynamic probabilistic logics as well. Finally, we discuss how our basic update rule can be parameterized for different ‘update policies’.

This paper reviews and examines interpretations of belief functions in the theory of rough sets with finite universe. The concept of standard rough set algebras is generalized in two directions. One is based on the use of non-equivalence relations. The other is based on relations over two universes, which leads to the notion of interval algebras. Pawlak rough set algebras may be used to interpret belief functions whose focal elements form a partition of the universe. Generalized rough set algebras using non-equivalence relations may be used to interpret belief functions which have less than |U | focal elements, where |U | is the cardinality of the universe U on which belief functions are defined. Interval algebras may be used to interpret any belief functions. 1.

In this paper we formulate the problem of inference under incomplete information in very general terms. This includes modelling the process responsible for the incompleteness, which we call the incompleteness process. We allow the process ’ behaviour to be partly unknown. Then we use Walley’s theory of coherent lower previsions, a generalisation of the Bayesian theory to imprecision, to derive the rule to update beliefs under incompleteness that logically follows from our assumptions, and that we call conservative inference rule. This rule has some remarkable properties: it is an abstract rule to update beliefs that can be applied in any situation or domain; it gives us the opportunity to be neither too optimistic nor too pessimistic about the incompleteness process, which is a necessary condition to draw reliable while strong enough conclusions; and it is a coherent rule, in the sense that it cannot lead to inconsistencies. We give examples to show how the new rule can be applied in expert systems, in parametric statistical inference, and in pattern classification, and discuss more generally the view of incompleteness processes defended here as well as some of its consequences. 1.

The emergence of robustness as an important consideration in Bayesian statistical models has led to a renewed interest in normative models of incomplete preferences represented by imprecise (set-valued) probabilities and utilities.

INTRODUCTION The best understood and most highly developed theory of uncertainty is Bayesian probability. There is a large literature on its foundations and there are many different justifications of the theory; however, all of these assume that for any proposition a, the beliefs in a and :a are strongly tied together. Without compelling justification, this assumption greatly restricts the type of information that can be satisfactorily represented, e.g., it makes it impossible to represent adequately partial information about an unknown chance distribution P such as 0:6 P(a) 0:8. The strict Bayesian requirement that an epistemic state be a single probability function seems unreasonable. A natural extension of the Bayesian theory is thus to allow sets of probability functions and to consider constraints and bounds on these, and to calculate s

this paper is to extend this investigation in several ways. The second contribution is to measure the relevance/utility of non-partitional questions, and show how di#erent proposals (using either protocols or likelihood functions) come down to the same thing

Currently, there is renewed interest in the problem, raised by Shafer in 1985, of updating probabilities when observations are incomplete (or setvalued) . This is a fundamental problem, and of particular interest for Bayesian networks. Recently, Gr unwald and Halpern have shown that commonly used updating strategies fail here, except under very special assumptions. We propose a new rule for updating probabilities with incomplete observations. Our approach is deliberately conservative: we make no or weak assumptions about the so-called incompleteness mechanism that produces incomplete observations. We model our ignorance about this mechanism by a vacuous lower prevision, a tool from the theory of imprecise probabilities, and we derive a new updating rule using coherence arguments. In general, our rule produces lower posterior probabilities, as well as partially determinate decisions.

Abstract—In multiagent interactions, such as e-commerce and file sharing, being able to accurately assess the trustworthiness of others is important for agents to protect themselves from losing utility. Focusing on rational agents in e-commerce, we prove that an agent’s discount factor (time preference of utility) is a direct measure of the agent’s trustworthiness for a set of reasonably general assumptions and definitions. We propose a general list of desiderata for trust systems and discuss how discount factors as trustworthiness meet these desiderata. We discuss how discount factors are a robust measure when entering commitments that exhibit moral hazards. Using an online market as a motivating example, we derive some analytical methods both for measuring discount factors and for aggregating the measurements. 1

A careful analysis of conditioning in the Sleeping Beauty problem is done, using the formal model for reasoning about knowledge and probability developed by Halpern and Tuttle. While the Sleeping Beauty problem has been viewed as revealing problems with conditioning in the presence of imperfect recall, the analysis done here reveals that the problems are not so much due to imperfect recall as to asynchrony. The implications of this analysis for van Fraassen’s Reflection Principle and Savage’s Sure-Thing Principle are considered. 1