Kolmogorov's axioms for probabilities with values in hyperbolic numbers

We introduce the notion of a probabilistic measure which takes values in hyperbolic numbers and which satisfies the system of axioms generalizing directly Kolmogorov's system of axioms. We show that this new measure verifies the usual properties of a probability; in particular, we treat the conditional hyperbolic probability and we prove the hyperbolic analogues of the multiplication theorem, of the law of total probability and of Bayes' theorem. Our probability may take values which are zero--divisors and we discuss carefully this peculiarity

Bayesian Decision Theory and Stochastic Independence

Stochastic independence has a complex status in probability theory. It is not part of the definition of a probability measure, but it is nonetheless an essential property for the mathematical development of this theory. Bayesian decision theorists such as Savage can be criticized for being silent about stochastic independence. From their current preference axioms, they can derive no more than the definitional properties of a probability measure. In a new framework of twofold uncertainty, we introduce preference axioms that entail not only these definitional properties, but also the stochastic independence of the two sources of uncertainty. This goes some way towards filling a curious lacuna in Bayesian decision theory

Probability functions in the context of signed involutive meadows

The Kolmogorov axioms for probability functions are placed in the context of signed meadows. A completeness theorem is stated and proven for the resulting equational theory of probability calculus. Elementary definitions of probability theory are restated in this framework.Comment: 20 pages, 6 tables, some minor errors are correcte

The World According to GARP* : Non-parametric Tests of Demand Theory and Rational Behavior

The purpose of this paper is twofold. We first point out that violation of rationality axioms (SARP, GARP, WARP) do not necessarily lead to a non-rational behavior. Second, our tests of axioms SARP, GARP and WARP over a Polish panel data (1987-90) show that over the 3630 households only 240 violate the three axioms. However these 240 violations are not caused by the non-respect of demand theory axioms but by the changing of preferences over the period. A logistic regression confirms the robustness of the test since the more the real expenditure increases in absolute value, the more the probability of violating the axioms increases (the respect of the axioms by the 3390 households is not due to an increase of the real expenditure). Moreover, changing in the composition of the family structure increases the probability of being inconsistent. It seems therefore that the 240 apparent violations are due to the appearance of new constraints, which increase the shadow prices of the goods. In order to explain these 240 households' preference changes, we build an econometric model of prices including an observed monetary component and an unobserved non-monetary component expressing the constraints faced by the agent. The estimation of this econometric model shows that the agents who apparently violate the axioms have these complete price changes superior to those of the agents who respect the axioms. Thus the agents who apparently violate the axioms faced during the period a change of their non-monetary resources and the appearance of new constraints.Rationality, GARP, Non-parametric tests, Shadow prices.

Quantum mechanics as a theory of probability

We develop and defend the thesis that the Hilbert space formalism of quantum mechanics is a new theory of probability. The theory, like its classical counterpart, consists of an algebra of events, and the probability measures defined on it. The construction proceeds in the following steps: (a) Axioms for the algebra of events are introduced following Birkhoff and von Neumann. All axioms, except the one that expresses the uncertainty principle, are shared with the classical event space. The only models for the set of axioms are lattices of subspaces of inner product spaces over a field K. (b) Another axiom due to Soler forces K to be the field of real, or complex numbers, or the quaternions. We suggest a probabilistic reading of Soler's axiom. (c) Gleason's theorem fully characterizes the probability measures on the algebra of events, so that Born's rule is derived. (d) Gleason's theorem is equivalent to the existence of a certain finite set of rays, with a particular orthogonality graph (Wondergraph). Consequently, all aspects of quantum probability can be derived from rational probability assignments to finite "quantum gambles". We apply the approach to the analysis of entanglement, Bell inequalities, and the quantum theory of macroscopic objects. We also discuss the relation of the present approach to quantum logic, realism and truth, and the measurement problem.Comment: 37 pages, 3 figures. Forthcoming in a Festschrift for Jeffrey Bub, ed. W. Demopoulos and the author, Springer (Kluwer): University of Western Ontario Series in Philosophy of Scienc

Attitude toward imprecise information

This paper presents an axiomatic model of decision making under uncertainty which incorporates objective but imprecise information. Information is assumed to take the form of a probability-possibility set, that is, a set $P$ of probability measures on the state space. The decision maker is told that the true probability law lies in $P$ and is assumed to rank pairs of the form $(P,f)$ where $f$ is an act mapping states into outcomes. The key representation result delivers maxmin expected utility where the min operator ranges over a set of probability priors --just as in the maxmin expected utility (MEU) representation result of \cite{GILB/SCHM/89}. However, unlike the MEU representation, the representation here also delivers a mapping, $\varphi$, which links the probability-possibility set, describing the available information, to the set of revealed priors. The mapping $\varphi$ is shown to represent the decision maker's attitude to imprecise information: under our axioms, the set of representation priors is constituted as a selection from the probability-possibility set. This allows both expected utility when the selected set is a singleton and extreme pessimism when the selected set is the same as the probability-possibility set, i.e. , $\varphi$ is the identity mapping. We define a notion of comparative imprecision aversion and show it is characterized by inclusion of the sets of revealed probability distributions, irrespective of the utility functions that capture risk attitude. We also identify an explicit attitude toward imprecision that underlies usual hedging axioms. Finally, we characterize, under extra axioms, a more specific functional form, in which the set of selected probability distributions is obtained by (i) solving for the ``mean value'' of the probability-possibility set, and (ii) shrinking the probability-possibility set toward the mean value to a degree determined by preferences.Imprecise information; imprecision aversion; multiple priors; Steiner point