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

On attitude polarization under Bayesian learning with non-additive beliefs

Ample psychological evidence suggests that people’s learning behavior is often prone to a "myside bias" or "irrational belief persistence" in contrast to learning behavior exclusively based on objective data. In the context of Bayesian learning such a bias may result in diverging posterior beliefs and attitude polarization even if agents receive identical information. Such patterns cannot be explained by the standard model of rational Bayesian learning that implies convergent beliefs. As our key contribution, we therefore develop formal models of Bayesian learning with psychological bias as alternatives to rational Bayesian learning. We derive conditions under which beliefs may diverge in the learning process despite the fact that all agents observe the same - arbitrarily large - sample, which is drawn from an "objective" i.i.d. process. Furthermore, one of our learning scenarios results in attitude polarization even in the case of common priors. Key to our approach is the assumption of ambiguous beliefs that are formalized as non-additive probability measures arising in Choquet expected utility theory. As a specific feature of our approach, our models of Bayesian learning with psychological bias reduce to rational Bayesian learning in the absence of ambiguity.Non-additive Probability Measures, Choquet Expected Utility Theory, Bayesian Learning, Bounded Rationality

Generating ambiguity in the laboratory

This article develops a method for drawing samples from which it is impossible to infer any quantile or moment of the underlying distribution. The method provides researchers with a way to give subjects the experience of ambiguity. In any experiment, learning the distribution from experience is impossible for the subjects, essentially because it is impossible for the experimenter. We describe our method mathematically, illustrate it in simulations, and then test it in a laboratory experiment. Our technique does not withhold sampling information, does not assume that the subject is incapable of making statistical inferences, is replicable across experiments, and requires no special apparatus. We compare our method to the techniques used in related experiments that attempt to produce an ambiguous experience for the subjects.ambiguity; Ellsberg; Knightian uncertainty; laboratory experiments; ignorance; vagueness JEL Classications: C90; C91; C92; D80; D81

Beyond Earthquakes: The New Directions of Expected Utility Theory

Over the past two decades or so, an enormous amount of work has been done to improve the Expected Utility model. Two areas have attracted major attention: the possibility of describing unforeseen contingencies and the need to accommodate the kind of behavior referred to in Ellsberg’s paradox. This essay surveys both.

Heat and Gravitation. III. Mixtures

The standard treatment of relativistic thermodynamics does not allow for a systematic treatment of mixtures. It is proposed that a formulation of thermodynamics as an action principle may be a suitable approach to adopt for a new investigation. This third paper of the series applies the action principle to a study of mixtures of ideal gases. The action for a mixture of ideal gases is the sum of the actions for the components, with an entropy that, in the absence of gravity, is determined by the Gibbs-Dalton hypothesis. Chemical reactions such as hydrogen dissociation are studied, with results that include the Saha equation and that are more complete than traditional treatments, especially so when gravitational effects are included. A mixture of two ideal gases is a system with two degrees of freedom and consequently it exhibits two kinds of sound. In the presence of gravity the Gibbs-Dalton hypothesis is modified to get results that agree with observation. The possibility of a parallel treatment of real gases is illustrated by an application to van der Waals gases. The overall conclusion is that experimental results serve to pin down the lagrangian in a very efficient manner. This leads to a convenient theoretical framework in which many dynamical problems can be studied.Comment: 33 pages, plain te

Nonlinear equation for curved stationary flames

A nonlinear equation describing curved stationary flames with arbitrary gas expansion $\theta = \rho_{{\rm fuel}}/\rho_{{\rm burnt}}$, subject to the Landau-Darrieus instability, is obtained in a closed form without an assumption of weak nonlinearity. It is proved that in the scope of the asymptotic expansion for $\theta \to 1,$ the new equation gives the true solution to the problem of stationary flame propagation with the accuracy of the sixth order in $\theta - 1.$ In particular, it reproduces the stationary version of the well-known Sivashinsky equation at the second order corresponding to the approximation of zero vorticity production. At higher orders, the new equation describes influence of the vorticity drift behind the flame front on the front structure. Its asymptotic expansion is carried out explicitly, and the resulting equation is solved analytically at the third order. For arbitrary values of $\theta,$ the highly nonlinear regime of fast flow burning is investigated, for which case a large flame velocity expansion of the nonlinear equation is proposed.Comment: 29 pages 4 figures LaTe

The dual process account of reasoning: historical roots, problems and perspectives.

Despite the great effort that has been dedicated to the attempt to redefine expected utility theory on the grounds of new assumptions, modifying or moderating some axioms, none of the alternative theories propounded so far had a statistical confirmation over the full domain of applicability. Moreover, the discrepancy between prescriptions and behaviors is not limited to expected utility theory. In two other fundamental fields, probability and logic, substantial evidence shows that human activities deviate from the prescriptions of the theoretical models. The paper suggests that the discrepancy cannot be ascribed to an imperfect axiomatic description of human choice, but to some more general features of human reasoning and assumes the “dual-process account of reasoning” as a promising explanatory key. This line of thought is based on the distinction between the process of deliberate reasoning and that of intuition; where in a first approximation, “intuition” denotes a mental activity largely automatized and inaccessible from conscious mental activity. The analysis of the interactions between these two processes provides the basis for explaining the persistence of the gap between normative and behavioral patterns. This view will be explored in the following pages: central consideration will be given to the problem of the interactions between rationality and intuition, and the correlated “modularity” of the thought.