A betting interpretation for probabilities and Dempster-Shafer degrees of belief

There are at least two ways to interpret numerical degrees of belief in terms of betting: (1) you can offer to bet at the odds defined by the degrees of belief, or (2) you can judge that a strategy for taking advantage of such betting offers will not multiply the capital it risks by a large factor. Both interpretations can be applied to ordinary additive probabilities and used to justify updating by conditioning. Only the second can be applied to Dempster-Shafer degrees of belief and used to justify Dempster's rule of combination.Comment: 20 page

A mathematical theory of evidence for G.L.S. Shackle

Evidence Theory is a branch of mathematics that concerns the combination of empirical evidence in an individual's mind in order to construct a coherent picture of reality. Designed to deal with unexpected empirical evidence suggesting new possibilities, evidence theory has a lot in common with Shackle's idea of decision-making as a creative act. This essay investigates this connection in detail, pointing to the usefulness of evidence theory to formalise and extend Shackle's decision theory. In order to ease a proper framing of the issues involved, evidence theory is not only compared with Shackle's ideas but also with additive and sub-additive probability theories. Furthermore, the presentation of evidence theory does not refer to the original version only, but takes account of its most recent developments, too.

ON THE RATIONAL SCOPE OF PROBABILISTIC RULE-BASED INFERENCE SYSTEMS

Belief updating schemes in artificial intelligence may be viewed as three dimensional languages, consisting of a syntax (e.g. probabilities or certainty factors), a calculus (e.g. Bayesian or CF combination rules), and a semantics (i.e. cognitive interpretations of competing formalisms). This paper studies the rational scope of those languages on the syntax and calculus grounds. In particular, the paper presents an endomorphism theorem which highlights the limitations imposed by the conditional independence assumptions implicit in the CF calculus. Implications of the theorem to the relationship between the CF and the Bayesian languages and the Dempster-Shafer theory of evidence are presented. The paper concludes with a discussion of some implications on rule-based knowledge engineering in uncertain domains.Information Systems Working Papers Serie

Conditional Belief Structures

The mathematical theory of evidence (Shafer et al. [9]) has recently found much interest as an approach to treat uncertainty in expert and knowledge-based systems. Although the theory is very promising, there are not yet many practical applications. Modeling practice has still to be developed. This is a crucial task in view of facilitating the application of evidential modeling. It is the aim of this paper to discuss an important element of evidential modeling-conditional belief-within the scope of the mathematical theory of evidenc

Decision making with belief functions: Compatibility and incompatibility with the sure-thing principle

This article studies situations in which information is ambiguous and only part of it can be probabilized. It is shown that the information can be modeled through belief functions if and only if the nonprobabilizable information is subject to the principles of complete ignorance. Next the representability of decisions by belief functions on outcomes is justified by means of a neutrality axiom. The natural weakening of Savage's sure-thing principle to unambiguous events is examined and its implications for decision making are identified

COMPARING THE VALIDITY OF ALTERNATIVE BELIEF LANGUAGES: AN EXPERIMENTAL APPROACH

The problem of modeling uncertainty and inexact reasoning in rule-based expert systems is challenging on nonnative as well on cognitive grounds. First, the modular structure of the rule-based architecture does not lend itself to standard Bayesian inference techniques. Second, there is no consensus on how to model human (expert) judgement under uncertainty. These factors have led to a proliferation of quasi-probabilistic belief calculi which are widely-used in practice. This paper investigates the descriptive and external validity of three well-known "belief languages:" the Bayesian, ad-hoc Bayesian, and the certainty factors languages. These models are implemented in many commercial expert system shells, and their validity is clearly an important issue for users and designers of expert systems. The methodology consists of a controlled, within-subject experiment designed to measure the relative performance of alternative belief languages. The experiment pits the judgement of human experts with the recommendations generated by their simulated expert systems, each using a different belief language. Special emphasis is given to the general issues of validating belief languages and expert systems at large.Information Systems Working Papers Serie

PROLOG META-INTERPRETERS FOR RULE-BASED INFERENCE UNDER UNCERTAINTY

Uncertain facts and inexact rules can be represented and processed in standard Prolog through meta-interpretation. This requires the specification of appropriate parsers and belief calculi. We present a meta-interpreter that takes a rule-based belief calculus as an external variable. The certainty-factors calculus and a heuristic Bayesian belief-update model are then implemented as stand-alone Prolog predicates. These, in turn, are bound to the meta-interpreter environment through second-order programming. The resulting system is a powerful experimental tool which enables inquiry into the impact of various designs of belief calculi on the external validity of expert systems. The paper also demonstrates the (well-known) role of Prolog meta-interpreters in building expert system shells.Information Systems Working Papers Serie

Consumer Weighting of Hedonic and Utilitarian Dimensions ACRoss Judgments

This research examined how individuals trade-off between utilitarian and hedonic dimensions across different judgments: choice, pricing and liking. When attribute trade-offs were large, utilitarian attributes were weighted more in choice than in pricing, or liking judgments, consistent with the prominence hypothesis. However, for small attribute trade-offs individuals placed greater weights on utilitarian attributes in liking than in choice, or pricing. Although decisions were largely driven by high utilitarian values, the size of the trade-off impacted the weights assigned to hedonic attributes in a joint evaluation context. Hence, smaller trade-offs resulted in increased weighting of hedonic attributes in choice and pricing

La significación como extensión del heurístico de representatividad. Dos ejemplos

Two problems designed by Tversky and Kahneman (1973 and 1982) were administered to identify the representativeness heuristic. One of the problems, referred to as the conjunction fallacy (PFC), provides a description of a fictional character who must choose the more likely of two options: one referring to a single event and the other to the conjunction of that event and another. Qualitative information presented induces choice not altogether in keeping with probabilistic rules. The other problem (PP) requires assigning degrees of likelihood to two professions for one single fictional character by means of relevant qualitative and quantitative information. Objective: This study assesses the effect of meaning on the answers provided to the problems. Meaning is hypothesized to outweigh the representativeness heuristic and to be strongly linked to semantic and linguistic problem presentation and the information provided. It is also postulated that PFC induces the use of meaning more heavily than PP. Method: The sample was selected on accessibility grounds and consisted of young people entering national universities in the Greater Buenos Aires, coming from middle-income public and private high schools and without previous training in probability. Results and discussion: The difference found between the proportion of responses attributed to meaning between PFC (94%) and PP (77%) is consistent with the hypotheses.Se administraron dos problemas diseñados por Tversky y Kahneman (1973 y 1982) para identificar el Heurístico de Representatividad. Uno identificado como Problema de la Falacia de la Conjunción (PFC). Se brinda una descripción de un personaje ficticio y se debe elegir entre dos opciones según se la considere más probable; una de ellas referida a un suceso único y la otra a la conjunción de ese suceso con otro. La información cualitativa presentada induce la elección de las opciones en un sentido que se aleja de las normas probabilísticas. El otro problema requiere asignar mayor o menor probabilidad a dos profesiones de otro personaje ficticio para lo cual se brinda información cualitativa y cuantitativa pertinente (PP). Objetivo: Se estudia el efecto de la significación sobre las respuestas a dichos problemas. Se plantean como hipótesis que la significación trasciende a la aparición del heurístico de representatividad y está fuertemente vinculada al planteo lingüístico-semántico del problema y a la informaciónpresentada. También se postula que el relato brindado en el PFC es más inductor hacia la utilización de la significación que el presentado para el PP. Metodología: La muestra fue seleccionada por accesibilidad y estuvo constituida por jóvenes ingresantes a universidades nacionales del Gran Buenos Aires, provenientes de colegios públicos y privados de nivel socioeconómico medio sin formación previa en probabilidades. Resultados y discusión: La diferencia encontrada entre las proporciones de respuestas atribuibles a la significación entre el PFC (94%) y el PP (77%) concuerda con las hipótesis planteadas