76,112 research outputs found
The Information Geometry of Space and Time
Is the geometry of space a macroscopic manifestation of an underlying
microscopic statistical structure? Is geometrodynamics - the theory of gravity
- derivable from general principles of inductive inference? Tentative answers
are suggested by a model of geometrodynamics based on the statistical concepts
of entropy, information geometry, and entropic dynamics. The model shows
remarkable similarities with the 3+1 formulation of general relativity. For
example, the dynamical degrees of freedom are those that specify the conformal
geometry of space; there is a gauge symmetry under 3d diffeomorphisms; there is
no reference to an external time; and the theory is time reversible. There is,
in adition, a gauge symmetry under scale transformations. I conjecture that
under a suitable choice of gauge one can recover the usual notion of a
relativistic space-time.Comment: Presented at the 25th International Workshop on Maximum Entropy and
Bayesian Methods in Science and Engineering (San Jose, California, August.
2005). The revised version 2 contains minor revisions of language and
punctuatio
Towards a Statistical Geometrodynamics
Can the spatial distance between two identical particles be explained in
terms of the extent that one can be distinguished from the other? Is the
geometry of space a macroscopic manifestation of an underlying microscopic
statistical structure? Is geometrodynamics derivable from general principles of
inductive inference? Tentative answers are suggested by a model of
geometrodynamics based on the statistical concepts of entropy, information
geometry, and entropic dynamics.Comment: Invited talk at the Decoherence, Information, Entropy, and Complexity
Workshop, DICE02, September 2000, Piombino, Ital
Relational Entropic Dynamics of Particles
The general framework of entropic dynamics is used to formulate a relational
quantum dynamics. The main new idea is to use tools of information geometry to
develop an entropic measure of the mismatch between successive configurations
of a system. This leads to an entropic version of the classical best matching
technique developed by J. Barbour and collaborators. The procedure is
illustrated in the simple case of a system of N particles with global
translational symmetry. The generalization to other symmetries whether global
(rotational invariance) or local (gauge invariance) is straightforward. The
entropic best matching allows a quantum implementation Mach's principles of
spatial and temporal relationalism and provides the foundation for a method of
handling gauge theories in an informational framework.Comment: Presented at MaxEnt 2015, the 35th International Workshop on Bayesian
Inference and Maximum Entropy Methods in Science and Engineering (July
19--24, 2015, Potsdam NY, USA
A Covariant Approach to Entropic Dynamics
Entropic Dynamics (ED) is a framework for constructing dynamical theories of
inference using the tools of inductive reasoning. A central feature of the ED
framework is the special focus placed on time. In previous work a global
entropic time was used to derive a quantum theory of relativistic scalar
fields. This theory, however, suffered from a lack of explicit or manifest
Lorentz symmetry. In this paper we explore an alternative formulation in which
the relativistic aspects of the theory are manifest.
The approach we pursue here is inspired by the works of Dirac, Kuchar, and
Teitelboim in their development of covariant Hamiltonian methods. The key
ingredient here is the adoption of a local notion of time, which we call
entropic time. This construction allows the expression of arbitrary notion of
simultaneity, in accord with relativity. In order to ensure, however, that this
local time dynamics is compatible with the background spacetime we must impose
a set of Poisson bracket constraints; these constraints themselves result from
requiring the dynamcics to be path independent, in the sense of Teitelboim and
Kuchar.Comment: An extended version of work presented at MaxEnt 2016, the 36th
International Workshop on Bayesian Inference and Maximum Entropy Methods in
Science and Engineering; July 10-15 2016, Ghent, Belgiu
Group Analysis of Self-organizing Maps based on Functional MRI using Restricted Frechet Means
Studies of functional MRI data are increasingly concerned with the estimation
of differences in spatio-temporal networks across groups of subjects or
experimental conditions. Unsupervised clustering and independent component
analysis (ICA) have been used to identify such spatio-temporal networks. While
these approaches have been useful for estimating these networks at the
subject-level, comparisons over groups or experimental conditions require
further methodological development. In this paper, we tackle this problem by
showing how self-organizing maps (SOMs) can be compared within a Frechean
inferential framework. Here, we summarize the mean SOM in each group as a
Frechet mean with respect to a metric on the space of SOMs. We consider the use
of different metrics, and introduce two extensions of the classical sum of
minimum distance (SMD) between two SOMs, which take into account the
spatio-temporal pattern of the fMRI data. The validity of these methods is
illustrated on synthetic data. Through these simulations, we show that the
three metrics of interest behave as expected, in the sense that the ones
capturing temporal, spatial and spatio-temporal aspects of the SOMs are more
likely to reach significance under simulated scenarios characterized by
temporal, spatial and spatio-temporal differences, respectively. In addition, a
re-analysis of a classical experiment on visually-triggered emotions
demonstrates the usefulness of this methodology. In this study, the
multivariate functional patterns typical of the subjects exposed to pleasant
and unpleasant stimuli are found to be more similar than the ones of the
subjects exposed to emotionally neutral stimuli. Taken together, these results
indicate that our proposed methods can cast new light on existing data by
adopting a global analytical perspective on functional MRI paradigms.Comment: 23 pages, 5 figures, 4 tables. Submitted to Neuroimag
Entropic Dynamics
Entropic Dynamics is a framework in which dynamical laws are derived as an
application of entropic methods of inference. No underlying action principle is
postulated. Instead, the dynamics is driven by entropy subject to the
constraints appropriate to the problem at hand. In this paper we review three
examples of entropic dynamics. First we tackle the simpler case of a standard
diffusion process which allows us to address the central issue of the nature of
time. Then we show that imposing the additional constraint that the dynamics be
non-dissipative leads to Hamiltonian dynamics. Finally, considerations from
information geometry naturally lead to the type of Hamiltonian that describes
quantum theory.Comment: Invited contribution to the Entropy special volume on Dynamical
Equations and Causal Structures from Observation
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