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Symmetry
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Symmetry in Probability and Inference
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Symmetry in Probability and Inference

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: closed (30 June 2016) | Viewed by 31444

Special Issue Editor

Prof. Dr. Marlos Viana

E-Mail Website
Guest Editor
University of Illinois at Chicago (Retd.), Chicago, IL 60612, USA
Interests: symmetry-derived algebraic methods for data analysis; symmetry studies; applications of algebraic methods to statistics and probability with applications in physics (optics); molecular biology (symbolic sequences)

Special Issue Information

Dear Colleagues,

Papers should address any aspects of symmetry arguments in probability and statistical inference, such as, but not limited to: Constructive rules of probability and inference derived from symmetry arguments; relative probabilities; symmetric probability measures, symmetry in probability distributions, symmetry-related arguments in entropy (probabilistic) laws; epistemic probabilities and symmetry principles, symmetry arguments in the cognitive foundations of probability, statistical inference under symmetry, quantum statistical inference, asymmetric inference (in Markov processes), exchangeability and symmetry. Group-theoretic approaches to probability and inference, including those discussing aspects of symmetry invariance derived from symmetry arguments will be considered. Papers discussing covariance structures derived from symmetry arguments, for example, will also be considered. Annotated reviews may also be considered.

Authors may benefit from clearly distinguishing papers (of lesser interest) that apply statistical methods to assess simple parametric hypotheses of symmetry from papers (of higher interest) that, instead, propose new statistical models amenable to classes of symmetries.

All papers should explicitly discuss the role of symmetry in their area of research, making it of interest and accessibility to all readership of Symmetry.

The guest editor encourages all authors, if in doubt, to contact him [email protected] with a brief white paper on their topic.

Dr. Marlos Viana
Guest Editor

Published Papers (6 papers)

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251 KiB  
Article
Symmetry and Evidential Support
by Michael G. Titelbaum
Symmetry 2011, 3(3), 680-698; https://doi.org/10.3390/sym3030680 - 16 Sep 2011
Cited by 3 | Viewed by 4670
Abstract
This article proves that formal theories of evidential favoring must fail because they are inevitably language dependent. I begin by describing Carnap’s early confirmation theories to show how language dependence problems (like Goodman’s grue problem) arise. I then generalize to showthat any formal [...] Read more.
This article proves that formal theories of evidential favoring must fail because they are inevitably language dependent. I begin by describing Carnap’s early confirmation theories to show how language dependence problems (like Goodman’s grue problem) arise. I then generalize to showthat any formal favoring theory satisfying minimal plausible conditions will yield different judgments about the same evidence and hypothesis when they are expressed in alternate languages. This does not just indict formal theories of favoring; it also shows that something beyond our evidence must be invoked to substantively favor one hypothesis over another. Full article
(This article belongs to the Special Issue Symmetry in Probability and Inference)
449 KiB  
Article
Lattices of Graphical Gaussian Models with Symmetries
by Helene Gehrmann
Symmetry 2011, 3(3), 653-679; https://doi.org/10.3390/sym3030653 - 07 Sep 2011
Cited by 8 | Viewed by 4683
Abstract
In order to make graphical Gaussian models a viable modelling tool when the number of variables outgrows the number of observations, [1] introduced model classes which place equality restrictions on concentrations or partial correlations. The models can be represented by vertex and edge [...] Read more.
In order to make graphical Gaussian models a viable modelling tool when the number of variables outgrows the number of observations, [1] introduced model classes which place equality restrictions on concentrations or partial correlations. The models can be represented by vertex and edge coloured graphs. The need for model selection methods makes it imperative to understand the structure of model classes. We identify four model classes that form complete lattices of models with respect to model inclusion, which qualifies them for an Edwards–Havránek model selection procedure [2]. Two classes turn out most suitable for a corresponding model search. We obtain an explicit search algorithm for one of them and provide a model search example for the other. Full article
(This article belongs to the Special Issue Symmetry in Probability and Inference)
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278 KiB  
Article
Symmetry and the Brown-Freiling Refutation of the Continuum Hypothesis
by Paul Bartha
Symmetry 2011, 3(3), 636-652; https://doi.org/10.3390/sym3030636 - 06 Sep 2011
Viewed by 5421
Abstract
Freiling [1] and Brown [2] have put forward a probabilistic reductio argument intended to refute the Continuum Hypothesis. The argument relies heavily upon intuitions about symmetry in a particular scenario. This paper argues that the argument fails, but is still of interest for [...] Read more.
Freiling [1] and Brown [2] have put forward a probabilistic reductio argument intended to refute the Continuum Hypothesis. The argument relies heavily upon intuitions about symmetry in a particular scenario. This paper argues that the argument fails, but is still of interest for two reasons. First, the failure is unusual in that the symmetry intuitions are demonstrably coherent, even though other constraints make it impossible to find a probability model for the scenario. Second, the best probability models have properties analogous to non-conglomerability, motivating a proposed extension of that concept (and corresponding limits on Bayesian conditionalization). Full article
(This article belongs to the Special Issue Symmetry in Probability and Inference)
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303 KiB  
Article
Symmetry, Invariance and Ontology in Physics and Statistics
by Julio Michael Stern
Symmetry 2011, 3(3), 611-635; https://doi.org/10.3390/sym3030611 - 01 Sep 2011
Cited by 23 | Viewed by 6697
Abstract
This paper has three main objectives: (a) Discuss the formal analogy between some important symmetry-invariance arguments used in physics, probability and statistics. Specifically, we will focus on Noether’s theorem in physics, the maximum entropy principle in probability theory, and de Finetti-type theorems in [...] Read more.
This paper has three main objectives: (a) Discuss the formal analogy between some important symmetry-invariance arguments used in physics, probability and statistics. Specifically, we will focus on Noether’s theorem in physics, the maximum entropy principle in probability theory, and de Finetti-type theorems in Bayesian statistics; (b) Discuss the epistemological and ontological implications of these theorems, as they are interpreted in physics and statistics. Specifically, we will focus on the positivist (in physics) or subjective (in statistics) interpretations vs. objective interpretations that are suggested by symmetry and invariance arguments; (c) Introduce the cognitive constructivism epistemological framework as a solution that overcomes the realism-subjectivism dilemma and its pitfalls. The work of the physicist and philosopher Max Born will be particularly important in our discussion. Full article
(This article belongs to the Special Issue Symmetry in Probability and Inference)
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227 KiB  
Article
High-Dimensional Random Matrices from the Classical Matrix Groups, and Generalized Hypergeometric Functions of Matrix Argument
by Donald St. P. Richards
Symmetry 2011, 3(3), 600-610; https://doi.org/10.3390/sym3030600 - 26 Aug 2011
Cited by 8 | Viewed by 4271
Abstract
Results from the theory of the generalized hypergeometric functions of matrix argument, and the related zonal polynomials, are used to develop a new approach to study the asymptotic distributions of linear functions of uniformly distributed random matrices from the classical compact matrix groups. [...] Read more.
Results from the theory of the generalized hypergeometric functions of matrix argument, and the related zonal polynomials, are used to develop a new approach to study the asymptotic distributions of linear functions of uniformly distributed random matrices from the classical compact matrix groups. In particular, we provide a new approach for proving the following result of D’Aristotile, Diaconis, and Newman: Let the random matrix Hn be uniformly distributed according to Haar measure on the group of n × n orthogonal matrices, and let An be a non-random n × n real matrix such that tr (A'nAn) = 1. Then, as n→∞, √n tr AnHn converges in distribution to the standard normal distribution. Full article
(This article belongs to the Special Issue Symmetry in Probability and Inference)
5373 KiB  
Article
Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas
by Michael D. Perlman and Jon A. Wellner
Symmetry 2011, 3(3), 574-599; https://doi.org/10.3390/sym3030574 - 23 Aug 2011
Cited by 7 | Viewed by 4841
Abstract
Do there exist circular and spherical copulas in ℝd? That is, do there exist circularly symmetric distributions on the unit disk in ℝ2 and spherically symmetric distributions on the unit ball in ℝd, d ≥ 3, whose one-dimensional [...] Read more.
Do there exist circular and spherical copulas in ℝd? That is, do there exist circularly symmetric distributions on the unit disk in ℝ2 and spherically symmetric distributions on the unit ball in ℝd, d ≥ 3, whose one-dimensional marginal distributions are uniform? The answer is yes for d = 2 and 3, where the circular and spherical copulas are unique and can be determined explicitly, but no for d ≥ 4. A one-parameter family of elliptical bivariate copulas is obtained from the unique circular copula in ℝ2 by oblique coordinate transformations. Copulas obtained by a non-linear transformation of a uniform distribution on the unit ball in ℝd are also described, and determined explicitly for d = 2. Full article
(This article belongs to the Special Issue Symmetry in Probability and Inference)
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