This chapter explores the topic of imprecise probabilities as it relates to model validation. IP is a family of formal methods that aim to provide a better representationRepresentation of severe uncertainty than is possible with standard probabilistic methods. Among the methods discussed here are using sets of probabilities to represent uncertainty, and using functions that do not satisfy the additvity property. We discuss the basics of IP, some examples of IP in computer simulation contexts, possible interpretations of the IP framework (...) and some conceptual problems for the approach. We conclude with a discussion of IP in the context of model validation. (shrink)

Quantum mechanical entangled configurations of particles that do not satisfy Bell’s inequalities, or equivalently, do not have a joint probability distribution, are familiar in the foundational literature of quantum mechanics. Nonexistence of a joint probability measure for the correlations predicted by quantum mechanics is itself equivalent to the nonexistence of local hidden variables that account for the correlations (for a proof of this equivalence, see Suppes and Zanotti, 1981). From a philosophical standpoint it is natural to ask what sort of (...) concept can be used to provide a “joint” analysis of such quantum correlations. In other areas of application of probability, similar but different problems arise. A typical example is the introduction of upper and lower probabilities in the theory of belief. A person may feel uncomfortable assigning a precise probability to the occurrence of rain tomorrow, but feel comfortable saying the probability should be greater than ½ and less than ⅞. Rather extensive statistical developments have occurred for this framework. A thorough treatment can be found in Walley (1991) and an earlier measurement-oriented development in Suppes (1974). It is important to note that this focus on beliefs, or related Bayesian ideas, is not concerned, as we are here, with the nonexistence of joint probability distributions. Yet earlier work with no relation to quantum mechanics, but focused on conditions for existence has been published by many people. For some of our own work on this topic, see Suppes and Zanotti (1989). Still, this earlier work naturally suggested the question of whether or not upper and lower measures could be used in quantum mechanics, as a generalization of.. (shrink)

Sorkin’s recent proposal for a realist interpretation of quantum theory, the anhomomorphic logic or coevent approach, is based on the idea of a “quantum measure” on the space of histories. This is a generalisation of the classical measure to one which admits pair-wise interference and satisfies a modified version of the Kolmogorov probability sum rule. In standard measure theory the measure on the base set Ω is normalised to one, which encodes the statement that “Ω happens”. Moreover, the Kolmogorov sum (...) rule implies that the measure of any subset A is strictly positive if and only if A cannot be covered by a countable collection of subsets of zero measure. In quantum measure theory on the other hand, simple examples suffice to demonstrate that this is no longer true. We propose an appropriate generalisation, the quantum cover, which in addition to being a cover of A, satisfies the property that if the quantum measure of A is non-zero then this is also the case for at least one of the elements in the cover. Our work implies a non-triviality result for the coevent interpretation for Ω of finite cardinality, and allows us to cast the Peres-Kochen-Specker theorem in terms of quantum covers. (shrink)

The aim of this paper is to state the single most powerful argument for use of a non-classical logic in quantum mechanics. In outline the argument is the following. The working logic of a science is the logic of the events and propositions to which probabilities are assigned. A probability should be assigned to every element of the algebra of events. In the case of quantum mechanics probabilities may be assigned to events but not, without restriction, to the conjunction of (...) two events. The conclusion is that the working logic of quantum mechanics is not classical. The nature of the logic that is appropriate for quantum mechanics is examined. (shrink)

We prove the existence of hidden variables, or, what we call generalized common causes, for finite sequences of pairwise correlated random variables that do not have a joint probability distribution. The hidden variables constructed have upper probability distributions that are nonmonotonic. The theorem applies directly to quantum mechanical correlations that do not satisfy the Bell inequalities.

In previous work, a non-standard theory of probability was formulated and used to systematize interference effects involving the simplest type of quantum systems. The main result here is a self-contained, non-trivial generalization of that theory to capture interference effects involving a much broader range of quantum systems. The discussion also focuses on interpretive matters having to do with the actual/virtual distinction, non-locality, and conditional probabilities.

A non-monotonic theory of probability is put forward and shown to have applicability in the quantum domain. It is obtained simply by replacing Kolmogorov's positivity axiom, which places the lower bound for probabilities at zero, with an axiom that reduces that lower bound to minus one. Kolmogorov's theory of probability is monotonic, meaning that the probability of A is less then or equal to that of B whenever A entails B. The new theory violates monotonicity, as its name suggests; yet, (...) many standard theorems are also theorems of the new theory since Kolmogorov's other axioms are retained. What is of particular interest is that the new theory can accommodate quantum phenomena (photon polarization experiments) while preserving Boolean operations, unlike Kolmogorov's theory. Although non-standard notions of probability have been discussed extensively in the physics literature, they have received very little attention in the philosophical literature. One likely explanation for that difference is that their applicability is typically demonstrated in esoteric settings that involve technical complications. That barrier is effectively removed for non-monotonic probability theory by providing it with a homely setting in the quantum domain. Although the initial steps taken in this paper are quite substantial, there is much else to be done, such as demonstrating the applicability of non-monotonic probability theory to other quantum systems and elaborating the interpretive framework that is provisionally put forward here. Such matters will be developed in other works. (shrink)

This article takes up a suggestion that the reason we cannot find certain hidden variable theories for quantum mechanics, as in Bell’s theorem, is that we require them to assign joint probability distributions on incompatible observables. These joint distributions are problematic because they are empirically meaningless on one standard interpretation of quantum mechanics. Some have proposed getting around this problem by using generalized probability spaces. I present a theorem to show a sense in which generalized probability spaces can’t serve as (...) hidden variable theories for quantum mechanics, so the proposal for getting around Bell’s theorem fails. 1 Introduction2 Bell’s Theorem and Classical Probability Spaces2.1 Bell’s derivation of the Bell inequalities2.2 Pitowsky’s derivation of the Bell inequalities3 Incompatible Observables4 Generalized Probability Spaces5 A ‘No-Go’ Theorem6 Conclusions. (shrink)

This book compares various approaches to the interpretation of quantum mechanics, in particular those which are related to the key words "the Copenhagen interpretation", "the antirealist view", "quantum logic" and "hidden variable theory". Using the concept of "correlation" carefully analyzed in the context of classical probability and in quantum theory, the author provides a framework to compare these approaches. He also develops an extension of probability theory to construct a local hidden variable theory. The book should be of interest for (...) physicists and philosophers of science interested in the foundations of quantum theory. (shrink)

A deterministic model that accounts for the statistical behavior of random samples of identical particles is presented. The model is based on some nonmeasurable distribution of spin values in all directions. The mathematical existence of such distributions is proved by set-theoretical techniques, and the relation between these distributions and observed frequencies is explored within an appropriate extension of probability theory. The relation between quantum mechanics and the model is specified. The model is shown to be consistent with known polarization phenomena (...) and the existence of macroscopic magnetism. Finally.. (shrink)

The actual/virtual distinction is used to give an alternative account of quantum interference by way of a new theory of probability. The new theory is obtained by changing one of the axioms of the canonical theory of probability while keeping the other axioms fixed. It is used to give an alternative account of constructive quantum interference in the two-slit experiment. The account crucially involves a distinction between actual and virtual probabilities. Although actual probabilities are operational and virtual probabilities are not, (...) there is a substantial connection between them. Virtual probabilities may be obtained indirectly from actual probabilities. In showing this, interpretive considerations are brought to bear including some having to do with quantum non-locality. Directions for future research are discussed in closing.La distinción actual/virtual se utiliza para dar una explicación alternativa de la interferencia cuántica a través de una nueva teoría de la probabilidad. La nueva teoría se ha obtenido cambiando uno de los axiomas de la teoría canónica de la probabilidad y manteniendo los otros axiomas fijos. Se usa para dar una explicación alternativa de la interferencia cuántica constructiva en el experimento de la doble rendija. El informe fundamentalmente implica una distinción entre las probabilidades actuales y virtuales. Aunque las probabilidades actuales son operativas y las probabilidades virtuales no, hay una relación sustancial entre ellas. Probabilidades virtuales pueden obtenerse indirectamente a partir de las probabilidades reales. Para demostrar esta interpretación hay que tener en cuenta las consideraciones de interpretación que se ejercen entre ellos y que tienen relación con la no-localidad cuántica. Directrices para futuras investigaciones se abordan al final del artículo. (shrink)

In this paper we analyze the existence of joint probabilities for the Bell-type and GHZ entangled states. We then propose the usage of nonmonotonic upper probabilities as a tool to derive consistent joint upper probabilities for the contextual hidden variables. Finally, we show that for the extreme example of no error, the GHZ state allows for the definition of a joint upper probability that is consistent with the strong correlations.