Conditioning in Chaotic Probabilities Interpreted as a Generalized Markov Chain

Abstract We propose a new definition for conditioning in the Chaotic Probability framework. We show that the Conditional Chaotic Probability model that we propose can be given the interpretation of a generalized Markov chain. Chaotic Probabilities were introduced by Fine et al. as an attempt to model chance phenomena with a usual set of measures M endowed with an objective, frequentist interpretation instead of a compound hypothesis or behavioral subjective one. We follow the presentation of the univariate case chaotic probability model and provide an instrumental interpretation of random process measures consistent with a conditional chaotic probability source, which can be used as a tool for simulation of our model. Given a finite time series, we also present a universal method for estimation of conditional chaotic probability models that is based on the analysis of the relative frequencies taken along a set of subsequences chosen by a given set of rules

Analysis of Conditional Expressions

What David Lewis proved in 1976 was stronger than he realized. Not only can no system of logic can have a conditional connective with non-trivial conditional probability, but also no probability space can have even a single non-trivial conditional event. However, Lewis' definition of conditional connective is flawed, and does not apply to his original target, the Stalnaker/Thomasson C2 logic. Lewis assumed a property which Stalnaker's system does not have - McGee's export-import law. Modal models of Stalnaker's C2 exist for every first-order model. Stalnaker's corner connectives, when interpreted as Lycan-style quantified conditionals, do have nontrivial conditional probability. Interpreting propositions as indicator functions instead of sets of possible worlds, the modern Kolmogorov theory of conditional expectation opens new possibilities for simultaneously modeling both objective and subjective probability as the expectation of truth. I use the new interpretation to defend Lycan's theory of conditionals against an objection from Dorothy Edgington

Conditional Density Operators and the Subjectivity of Quantum Operations

Assuming that quantum states, including pure states, represent subjective degrees of belief rather than objective properties of systems, the question of what other elements of the quantum formalism must also be taken as subjective is addressed. In particular, we ask this of the dynamical aspects of the formalism, such as Hamiltonians and unitary operators. Whilst some operations, such as the update maps corresponding to a complete projective measurement, must be subjective, the situation is not so clear in other cases. Here, it is argued that all trace preserving completely positive maps, including unitary operators, should be regarded as subjective, in the same sense as a classical conditional probability distribution. The argument is based on a reworking of the Choi-Jamiolkowski isomorphism in terms of "conditional" density operators and trace preserving completely positive maps, which mimics the relationship between conditional probabilities and stochastic maps in classical probability.Comment: 10 Pages, Work presented at "Foundations of Probability and Physics-4", Vaxjo University, June 4-9 200

The Minimal Modal Interpretation of Quantum Theory

We introduce a realist, unextravagant interpretation of quantum theory that builds on the existing physical structure of the theory and allows experiments to have definite outcomes, but leaves the theory's basic dynamical content essentially intact. Much as classical systems have specific states that evolve along definite trajectories through configuration spaces, the traditional formulation of quantum theory asserts that closed quantum systems have specific states that evolve unitarily along definite trajectories through Hilbert spaces, and our interpretation extends this intuitive picture of states and Hilbert-space trajectories to the case of open quantum systems as well. We provide independent justification for the partial-trace operation for density matrices, reformulate wave-function collapse in terms of an underlying interpolating dynamics, derive the Born rule from deeper principles, resolve several open questions regarding ontological stability and dynamics, address a number of familiar no-go theorems, and argue that our interpretation is ultimately compatible with Lorentz invariance. Along the way, we also investigate a number of unexplored features of quantum theory, including an interesting geometrical structure---which we call subsystem space---that we believe merits further study. We include an appendix that briefly reviews the traditional Copenhagen interpretation and the measurement problem of quantum theory, as well as the instrumentalist approach and a collection of foundational theorems not otherwise discussed in the main text.Comment: 73 pages + references, 9 figures; cosmetic changes, added figure, updated references, generalized conditional probabilities with attendant changes to the sections on the EPR-Bohm thought experiment and Lorentz invariance; for a concise summary, see the companion letter at arXiv:1405.675

Multiple perspectives on the concept of conditional probability

Conditional probability is a key to the subjectivist theory of probability; however, it plays a subsidiary role in the usual conception of probability where its counterpart, namely independence is of basic importance. The paper investigates these concepts from various perspectives in order to shed light on their multi-faceted character. We will include the mathematical, philosophical, and educational perspectives. Furthermore, we will inspect conditional probability from the corners of competing ideas and solving strategies. For the comprehension of conditional probability, a wider approach is urgently needed to overcome the well-known problems in learning the concepts, which seem nearly unaffected by teaching

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

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