THE TWO SCOPES OF FUZZY PROBABILITY THEORY

The aim of this work is to compare between what seems to be entirely different two highly developing “fuzzy probability” theories. The first theory had been developed firstly by statisticians and the other separately by physicists. We start by indicating the needs to develop such theories and what helped to develop each, then we will establish the basis of the two theories and illustrate that each indeed extends classical probability theory. Moreover, we will try to see whether or not any of the two theory can be embedded into the other

Classical Extensions, Classical Representations and Bayesian Updating in Quantum Mechanics

I review the formalism of classical extensions of quantum mechanics introduced by Beltrametti and Bugajski, and compare it to the classical representations discussed e.g. by Busch, Hellwig and Stulpe and recently used by Fuchs in his discussion of quantum mechanics in terms of standard quantum measurements. I treat the problem of finding Bayesian analogues of the state transition associated with measurement in the canonical classical extension as well as in the related 'uniform' classical representation. In the classical extension, the analogy is extremely good.Comment: 14 pages, presented at the conference 'Quantum Theory: Reconsideration of Foundations - 2', Vaexjoe, Sweden, June 200

The structure of classical extensions of quantum probability theory

On the basis of a suggestive definition of a classical extension of quantum mechanics in terms of statistical models, we prove that every such classical extension is essentially given by the so-called Misra–Bugajski reduction map. We consider how this map enables one to understand quantum mechanics as a reduced classical statistical theory on the projective Hilbert space as phase space and discuss features of the induced hidden-variable model. Moreover, some relevant technical results on the topology and Borel structure of the projective Hilbert space are reviewed

Contexts in Quantum Measurement Theory

State transformations in quantum mechanics are described by completely positive maps which are constructed from quantum channels. We call a finest sharp quantum channel a context. The result of a measurement depends on the context under which it is performed. Each context provides a viewpoint of the quantum system being measured. This gives only a partial picture of the system which may be distorted and in order to obtain a total accurate picture, various contexts need to be employed. We first discuss some basic definitions and results concerning quantum channels. We briefly describe the relationship between this work and ontological models that form the basis for contextuality studies. We then consider properties of channels and contexts. For example, we show that the set of sharp channels can be given a natural partial order in which contexts are the smallest elements. We also study properties of channel maps. The last section considers mutually unbiased contexts. These are related to mutually unbiased bases which have a large current literature. Finally, we connect them to completely random channel maps.Comment: 19 page

Just How Final are Today's Quantum Structures?

I present a selection of conceptual and mathematical problems in the foundations of modern physics as they derive from the title question. Contribution to a panel session, "Springer Forum: Quantum Structures -- Physical, Mathematical and Epistemological Problems", held at the Biannual Symposium of the International Quantum Structures Association, Liptovsky Jan, September 1998. To appear in journal: Soft Computing (2001).Comment: 3 pages, tcilate

Conditional probability of actually detecting a financial fraud - a neutrosophic extension to Benford's law

This study actually draws from and builds on an earlier paper (Kumar and Bhattacharya, 2002). Here we have basically added a neutrosophic dimension to the problem of determining the conditional probability that a financial fraud has been actually committed, given that no Type I error occurred while rejecting the null hypothesis H0: The observed first-digit frequencies approximate a Benford distribution; and accepting the alternative hypothesis H1: The observed first-digit frequencies do not approximate a Benford distribution. We have also suggested a conceptual model to implement such a neutrosophic fraud detection system.Comment: 9 page