Reasoning with imprecise probabilities

This special issue of the International Journal of Approximate Reasoning (IJAR) grew out of the 4th International Symposium on Imprecise Probabilities and Their Applications (ISIPTA’05), held in Pittsburgh, USA, in July 2005 (http://www.sipta.org/isipta05). The symposium was organized by Teddy Seidenfeld, Robert Nau, and Fabio G. Cozman, and brought together researchers from various branches interested in imprecision in probabilities. Research in artificial intelligence, economics, engineering, psychology, philosophy, statistics, and other fields was presented at the meeting, in a lively atmosphere that fostered communication and debate. Invited talks by Isaac Levi and Arthur Dempster enlightened the attendants, while tutorials by Gert de Cooman, Paolo Vicig, and Kurt Weichselberger introduced basic (and advanced) concepts; finally, the symposium ended with a workshop on financial risk assessment, organized by Teddy Seidenfeld

How much of commonsense and legal reasoning is formalizable? A review of conceptual obstacles

Fifty years of effort in artificial intelligence (AI) and the formalization of legal reasoning have produced both successes and failures. Considerable success in organizing and displaying evidence and its interrelationships has been accompanied by failure to achieve the original ambition of AI as applied to law: fully automated legal decision-making. The obstacles to formalizing legal reasoning have proved to be the same ones that make the formalization of commonsense reasoning so difficult, and are most evident where legal reasoning has to meld with the vast web of ordinary human knowledge of the world. Underlying many of the problems is the mismatch between the discreteness of symbol manipulation and the continuous nature of imprecise natural language, of degrees of similarity and analogy, and of probabilities

The Set Structure of Precision: Coherent Probabilities on Pre-Dynkin-Systems

In literature on imprecise probability little attention is paid to the fact that imprecise probabilities are precise on some events. We call these sets system of precision. We show that, under mild assumptions, the system of precision of a lower and upper probability form a so-called (pre-)Dynkin-system. Interestingly, there are several settings, ranging from machine learning on partial data over frequential probability theory to quantum probability theory and decision making under uncertainty, in which a priori the probabilities are only desired to be precise on a specific underlying set system. At the core of all of these settings lies the observation that precise beliefs, probabilities or frequencies on two events do not necessarily imply this precision to hold for the intersection of those events. Here, (pre-)Dynkin-systems have been adopted as systems of precision, too. We show that, under extendability conditions, those pre-Dynkin-systems equipped with probabilities can be embedded into algebras of sets. Surprisingly, the extendability conditions elaborated in a strand of work in quantum physics are equivalent to coherence in the sense of Walley (1991, Statistical reasoning with imprecise probabilities, p. 84). Thus, literature on probabilities on pre-Dynkin-systems gets linked to the literature on imprecise probability. Finally, we spell out a lattice duality which rigorously relates the system of precision to credal sets of probabilities. In particular, we provide a hitherto undescribed, parametrized family of coherent imprecise probabilities

Making decisions with evidential probability and objective Bayesian calibration inductive logics

Calibration inductive logics are based on accepting estimates of relative frequencies, which are used to generate imprecise probabilities. In turn, these imprecise probabilities are intended to guide beliefs and decisions — a process called “calibration”. Two prominent examples are Henry E. Kyburg's system of Evidential Probability and Jon Williamson's version of Objective Bayesianism. There are many unexplored questions about these logics. How well do they perform in the short-run? Under what circumstances do they do better or worse? What is their performance relative to traditional Bayesianism? In this article, we develop an agent-based model of a classic binomial decision problem, including players based on variations of Evidential Probability and Objective Bayesianism. We compare the performances of these players, including against a benchmark player who uses standard Bayesian inductive logic. We find that the calibrated players can match the performance of the Bayesian player, but only with particular acceptance thresholds and decision rules. Among other points, our discussion raises some challenges for characterising “cautious” reasoning using imprecise probabilities. Thus, we demonstrate a new way of systematically comparing imprecise probability systems, and we conclude that calibration inductive logics are surprisingly promising for making decisions

Empirical interpretation of imprecise probabilities

This paper investigates the possibility of a frequentist interpretation of imprecise probabilities, by generalizing the approach of Bernoulli’s Ars Conjectandi. That is, by studying, in the case of games of chance, under which assumptions imprecise probabilities can be satisfactorily estimated from data. In fact, estimability on the basis of finite amounts of data is a necessary condition for imprecise probabilities in order to have a clear empirical meaning. Unfortunately, imprecise probabilities can be estimated arbitrarily well from data only in very limited settings

The Goodman-Nguyen Relation within Imprecise Probability Theory

The Goodman-Nguyen relation is a partial order generalising the implication (inclusion) relation to conditional events. As such, with precise probabilities it both induces an agreeing probability ordering and is a key tool in a certain common extension problem. Most previous work involving this relation is concerned with either conditional event algebras or precise probabilities. We investigate here its role within imprecise probability theory, first in the framework of conditional events and then proposing a generalisation of the Goodman-Nguyen relation to conditional gambles. It turns out that this relation induces an agreeing ordering on coherent or C-convex conditional imprecise previsions. In a standard inferential problem with conditional events, it lets us determine the natural extension, as well as an upper extension. With conditional gambles, it is useful in deriving a number of inferential inequalities.Comment: Published version: http://www.sciencedirect.com/science/article/pii/S0888613X1400101

Perturbation bounds and degree of imprecision for uniquely convergent imprecise Markov chains

The effect of perturbations of parameters for uniquely convergent imprecise Markov chains is studied. We provide the maximal distance between the distributions of original and perturbed chain and maximal degree of imprecision, given the imprecision of the initial distribution. The bounds on the errors and degrees of imprecision are found for the distributions at finite time steps, and for the stationary distributions as well.Comment: 20 pages, 2 figure

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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