General Probabilistic Theories Research Articles

Algebraic properties of complex fuzzy events in classical and in quantum information systems

Baer ∗-semigroups are regarded as the main abstract structures for an algebraic analysis of complex fuzzy events in generalized probability theory. This assumption is verified in the case of classical probability theory in the framework of measure and integration theory. The corresponding fuzzy language is extended to the non-commutative probability theory based on operators in Hilbert space. Starting from a quantum information system a quantum probability space is constructed, which is naturally embedded in a classical information system. In this last both exact than fuzzy quantum events are represented as classical fuzzy events. Lastly, the classical fuzzy events which correspond to exact quantum events are characterized by some minimality properties.

Calibration-Based Empirical Probability

Probability forecasts for a sequence of uncertain events may be compared with the outcomes of those events by means of a natural criterion of empirical validity, calibration. It is shown that any two sequences of forecasts which both meet this criterion must be in asymptotic agreement. These agreed values can then be considered as correct objective probability forecasts for the particular sequence of outcome results obtained. However, the objective forecasts vary with the extent of the information taken into account when they are formulated. We thus obtain a general theory of empirical probability, relative to an information base. This theory does not require that such probabilities be interpreted in terms of repeated trials of the same event. Some implications of this theory are discussed.

Reality, locality, and probability

It is frequently argued that reality and locality are incompatible with the predictions of quantum mechanics. Various investigators have used this as evidence for the existence of hidden variables. However, Bell's inequalities seem to refute this possibility. Since the above arguments are made within the framework of conventional probability theory, we contend that an alternative solution can be found by an extension of this theory. Elaborating on some ideas of I. Pitowski, we show that within the framework of a generalized probability theory, reality, locality, hidden variables, and the predictions of quantum mechanics can be maintained together. Although our principal model in this work is a spin system, there are indications that this program can be extended to more general systems.

Nonspatial Candidate Characteristics and Electoral Competition

BASIC TO MODELS of electoral competition is the assumption that candidates adopt positions on policy issues as a means of attracting votes. However, candidates are also judged on the basis of human qualities and other attributes not related to the policies they espouse in a campaign. The spatial theory of electoral competition has, in the past, treated such attributes as part of the policy space over which candidates compete. However, certain difficulties attend such an interpretation. For example, it is difficult to view a candidate's personality as something which can be altered to please the voters. The mistakes that an incumbent has committed in office are not things he can erase to compete more effectively for votes. A candidate's religion cannot be abandoned because it is a political liability. In short, there are nonspatial attributes that affect voter evaluations of each candidate, which are beyond that candidate's immediate control. It is the purpose of this paper to incorporate these nonspatial attributes into the spatial model of electoral competition to show how the policy outcome of two candidate electoral competition is af-

A probabilistic theory of the coincidence method. I. Centrosymmetric space groups

A general probabilistic theory of sign coincidences is described which is valid in all the centrosymmetric space groups. The theory makes full use of space-group symmetry by means of suitable joint probability distribution functions. The phase relationships obtained are in some way space-group dependent: their estimation requires an appropriate use of space-group algebra.

Conditional expectations in generalized probability theory

Expectations have been considered as dual objects of instruments in several papers on generalized probability theory and quantum theory. Here, we relate a generalized conditional (GC) expectation to a given instrument and a given state by two requirements, which are analogous to the axioms by which the classical conditional expectation is related to a given sub-σ-algebra and a given probability measure. In examples we illustrate the close similarity of GC expectations with classical conditional expectations. Eventually, we introduce a rich class of quantum stochastic processes, which are Markovian.

Quantum logic and generalized probability theory

Quantum logic and generalized probability theory

Relevance weighting of search terms

AbstractThis paper examines statistical techniques for exploiting relevance information to weight search terms. These techniques are presented as a natural extension of weighting methods using information about the distribution of index terms in documents in general. A series of relevance weighting functions is derived and is justified by theoretical considerations. In particular, it is shown that specific weighted search methods are implied by a general probabilistic theory of retrieval. Different applications of relevance weighting are illustrated by experimental results for test collections.

A choice theory of spatial interaction

A general probabilistic theory is developed for modeling a variety of spatial interaction choices by individual behaving units. This theory, designated as the Choice Theory, is shown to be consistent with a wide class of empirical spatial interaction hypotheses, designated as Gravity Hypotheses. Moreover, it is shown that under certain structural conditions the Choice Theory completely characterizes this class of hypotheses. Finally, a number of explicit statistical procedures for testing the Choice Theory are developed and discussed.

Spectral Methods for a Generalized Probability Theory

Spectral Methods for a Generalized Probability Theory

Spectral methods for a generalized probability theory

Spectral methods for a generalized probability theory

A method for calculating the probability of detection for a coherent (AMTI) radar unit with phase cancellation

In this paper, general probability theory is used to evaluate the theoretical effectiveness of an Airborne Moving Target Indicator (AMTI) radar system using phase cancellation in the cancellation unit, this effectiveness being expressed in terms of a Probability of Detection. In developing the required expressions, a four-dimensional probability density function is first derived to relate the incoming signal at time ( l) with a signal that arrived (τ) seconds before. Using this function, the probability density functions for the primary units in the radar, namely, the receiver, the cancellation unit, the rectifier and the read-out scope (planned position indicator) are derived. Using the probability density function for the planned position indicator unit the expression for the probability of detection for the over-all system is obtained. The results show the probability of detection of the system to be a function of the signal to background noise (clutter) ratio, variance of the clutter power density spectrum and the pulse repetition frequency, and the results are given in a useful graph form.

Mathematical Statistics

PARTS III. and IV. of the twelfth volume of “Biometrika” contain papers of interest to all classes of statistician. Those especially attracted by work on the general mathematical theory of probability will welcome the continuation of Prof. Tchou-proff's paper on the “Expectation of the Moments of Frequency Distributions.” It will be agreed that the notation of mathematical expectation offers certain advantages over the iforms of expression more commonlv adopted in this; country, but Prof. Pearson does well to point out that the supposed fundamental distinction of method claimed by some Continental writers is nonexistent. He himself contributes a very interesting paper on. a method of generalising Tchebycheff's first theorem. He finds that the method of approximating to the limits of a probability is unlikely-to be of much practical value in the classes of function of usual occurrence. This result does not, of course, deprive Tchebycheff's work of its interest in permitting of the establishment of Bernoulli's theorem and of Poisson's generalisation of that theorem by elementary methods. Paper's of importance both to the student of theoretical -statistics and to the practical computer are those of Miss Pairman and Prof. Pearson on the correction of the moment coefficients in limited range frequency distributions, of “Student” on-deviations from the Poisson limit to the binomial in actual data, and an editorial,. entitled “Peceavimus,” correcting errors in various published formulae.

VII. On the theory of local probability, applied to straight lines drawn at random in a plane; the methods used being also extended to the proof of certain new theorems in the integral calculus

1. The new Theory of Local or Geometrical Probability, so far as it is known, seems to present, in a remarkable degree, the same distinguishing features which characterize those portions of the general Theory of Probability which we owe to the great philosophers of the past generation. The rigorous precision, as well as the extreme beauty of the methods and results, the extent of the demands made on our mathematical resources, even by cases apparently of the simplest kind, the subtlety and delicacy of the reasoning, which seem peculiar to that wonderful application of modern analysis— ce calcul delicat , as it has been aptly described by Laplace—reappear, under new forms, in this, its latest development. The first trace which we can discover of the Theory of Local Probability seems to be the celebrated problem of Buffon, the great naturalist— a given rod being placed at random on a space ruled with equidistant parallel lines, to find the chance of its crossing one of the lines. Although the subject was noticed so early, and though Buffon’s and one or two similar questions have been considered by Laplace, no real attention seems to have been bestowed upon it till within the last few years, when this new field of research has been entered upon by several English mathematicians, among whom the names of Sylvester and Woolhouse are particularly distinguished. It is true that in a few cases differences of opinion have arisen as to the principles, and discordant results have been arrived at, as in the now celebrated three-point problem, by Mr. Woolhouse, and the four-point problem of Professor Sylvester; but all feel that this arises, not from any inherent ambiguity in the subject matter, but from the weakness of the instrument employed; our undisciplined conceptions of a novel subject requiring to be repeatedly and patiently reviewed, tested, and corrected by the light of experience and comparison, before they are purged from all latent error. The object of the present paper is, principally, the application of the Theory of Probability to straight lines drawn at random in a plane; a branch of the subject which has not yet been investigated. It will be necessary to begin by some remarks on the general principles of Local Probability. Some portion of what follows I have already given elsewhere.