41,995 research outputs found
Measurement Invariance, Entropy, and Probability
We show that the natural scaling of measurement for a particular problem
defines the most likely probability distribution of observations taken from
that measurement scale. Our approach extends the method of maximum entropy to
use measurement scale as a type of information constraint. We argue that a very
common measurement scale is linear at small magnitudes grading into logarithmic
at large magnitudes, leading to observations that often follow Student's
probability distribution which has a Gaussian shape for small fluctuations from
the mean and a power law shape for large fluctuations from the mean. An inverse
scaling often arises in which measures naturally grade from logarithmic to
linear as one moves from small to large magnitudes, leading to observations
that often follow a gamma probability distribution. A gamma distribution has a
power law shape for small magnitudes and an exponential shape for large
magnitudes. The two measurement scales are natural inverses connected by the
Laplace integral transform. This inversion connects the two major scaling
patterns commonly found in nature. We also show that superstatistics is a
special case of an integral transform, and thus can be understood as a
particular way in which to change the scale of measurement. Incorporating
information about measurement scale into maximum entropy provides a general
approach to the relations between measurement, information and probability
Statistical mechanics of two-dimensional point vortices: relaxation equations and strong mixing limit
We complete the literature on the statistical mechanics of point vortices in
two-dimensional hydrodynamics. Using a maximum entropy principle, we determine
the multi-species Boltzmann-Poisson equation and establish a form of virial
theorem. Using a maximum entropy production principle (MEPP), we derive a set
of relaxation equations towards statistical equilibrium. These relaxation
equations can be used as a numerical algorithm to compute the maximum entropy
state. We mention the analogies with the Fokker-Planck equations derived by
Debye and H\"uckel for electrolytes. We then consider the limit of strong
mixing (or low energy). To leading order, the relationship between the
vorticity and the stream function at equilibrium is linear and the maximization
of the entropy becomes equivalent to the minimization of the enstrophy. This
expansion is similar to the Debye-H\"uckel approximation for electrolytes,
except that the temperature is negative instead of positive so that the
effective interaction between like-sign vortices is attractive instead of
repulsive. This leads to an organization at large scales presenting
geometry-induced phase transitions, instead of Debye shielding. We compare the
results obtained with point vortices to those obtained in the context of the
statistical mechanics of continuous vorticity fields described by the
Miller-Robert-Sommeria (MRS) theory. At linear order, we get the same results
but differences appear at the next order. In particular, the MRS theory
predicts a transition between sinh and tanh-like \omega-\psi relationships
depending on the sign of Ku-3 (where Ku is the Kurtosis) while there is no such
transition for point vortices which always show a sinh-like \omega-\psi
relationship. We derive the form of the relaxation equations in the strong
mixing limit and show that the enstrophy plays the role of a Lyapunov
functional
How to read probability distributions as statements about process
Probability distributions can be read as simple expressions of information.
Each continuous probability distribution describes how information changes with
magnitude. Once one learns to read a probability distribution as a measurement
scale of information, opportunities arise to understand the processes that
generate the commonly observed patterns. Probability expressions may be parsed
into four components: the dissipation of all information, except the
preservation of average values, taken over the measurement scale that relates
changes in observed values to changes in information, and the transformation
from the underlying scale on which information dissipates to alternative scales
on which probability pattern may be expressed. Information invariances set the
commonly observed measurement scales and the relations between them. In
particular, a measurement scale for information is defined by its invariance to
specific transformations of underlying values into measurable outputs.
Essentially all common distributions can be understood within this simple
framework of information invariance and measurement scale.Comment: v2: added table of contents, adjusted section numbers v3: minor
editing, updated referenc
Escort mean values and the characterization of power-law-decaying probability densities
Escort mean values (or -moments) constitute useful theoretical tools for
describing basic features of some probability densities such as those which
asymptotically decay like {\it power laws}. They naturally appear in the study
of many complex dynamical systems, particularly those obeying nonextensive
statistical mechanics, a current generalization of the Boltzmann-Gibbs theory.
They recover standard mean values (or moments) for . Here we discuss the
characterization of a (non-negative) probability density by a suitable set of
all its escort mean values together with the set of all associated normalizing
quantities, provided that all of them converge. This opens the door to a
natural extension of the well known characterization, for the instance,
of a distribution in terms of the standard moments, provided that {\it all} of
them have {\it finite} values. This question would be specially relevant in
connection with probability densities having {\it divergent} values for all
nonvanishing standard moments higher than a given one (e.g., probability
densities asymptotically decaying as power-laws), for which the standard
approach is not applicable. The Cauchy-Lorentz distribution, whose second and
higher even order moments diverge, constitutes a simple illustration of the
interest of this investigation. In this context, we also address some
mathematical subtleties with the aim of clarifying some aspects of an
interesting non-linear generalization of the Fourier Transform, namely, the
so-called -Fourier Transform.Comment: 20 pages (2 Appendices have been added
A parametrization of two-dimensional turbulence based on a maximum entropy production principle with a local conservation of energy
In the context of two-dimensional (2D) turbulence, we apply the maximum
entropy production principle (MEPP) by enforcing a local conservation of
energy. This leads to an equation for the vorticity distribution that conserves
all the Casimirs, the energy, and that increases monotonically the mixing
entropy (-theorem). Furthermore, the equation for the coarse-grained
vorticity dissipates monotonically all the generalized enstrophies. These
equations may provide a parametrization of 2D turbulence. They do not generally
relax towards the maximum entropy state. The vorticity current vanishes for any
steady state of the 2D Euler equation. Interestingly, the equation for the
coarse-grained vorticity obtained from the MEPP turns out to coincide, after
some algebraic manipulations, with the one obtained with the anticipated
vorticity method. This shows a connection between these two approaches when the
conservation of energy is treated locally. Furthermore, the newly derived
equation, which incorporates a diffusion term and a drift term, has a nice
physical interpretation in terms of a selective decay principle. This gives a
new light to both the MEPP and the anticipated vorticity method.Comment: To appear in the special IUTAM issue of Fluid Dynamics Research on
Vortex Dynamic
Revisiting maximum-a-posteriori estimation in log-concave models
Maximum-a-posteriori (MAP) estimation is the main Bayesian estimation
methodology in imaging sciences, where high dimensionality is often addressed
by using Bayesian models that are log-concave and whose posterior mode can be
computed efficiently by convex optimisation. Despite its success and wide
adoption, MAP estimation is not theoretically well understood yet. The
prevalent view in the community is that MAP estimation is not proper Bayesian
estimation in a decision-theoretic sense because it does not minimise a
meaningful expected loss function (unlike the minimum mean squared error (MMSE)
estimator that minimises the mean squared loss). This paper addresses this
theoretical gap by presenting a decision-theoretic derivation of MAP estimation
in Bayesian models that are log-concave. A main novelty is that our analysis is
based on differential geometry, and proceeds as follows. First, we use the
underlying convex geometry of the Bayesian model to induce a Riemannian
geometry on the parameter space. We then use differential geometry to identify
the so-called natural or canonical loss function to perform Bayesian point
estimation in that Riemannian manifold. For log-concave models, this canonical
loss is the Bregman divergence associated with the negative log posterior
density. We then show that the MAP estimator is the only Bayesian estimator
that minimises the expected canonical loss, and that the posterior mean or MMSE
estimator minimises the dual canonical loss. We also study the question of MAP
and MSSE estimation performance in large scales and establish a universal bound
on the expected canonical error as a function of dimension, offering new
insights into the good performance observed in convex problems. These results
provide a new understanding of MAP and MMSE estimation in log-concave settings,
and of the multiple roles that convex geometry plays in imaging problems.Comment: Accepted for publication in SIAM Imaging Science
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