2,449 research outputs found
Minimum Conditional Description Length Estimation for Markov Random Fields
In this paper we discuss a method, which we call Minimum Conditional
Description Length (MCDL), for estimating the parameters of a subset of sites
within a Markov random field. We assume that the edges are known for the entire
graph . Then, for a subset , we estimate the parameters
for nodes and edges in as well as for edges incident to a node in , by
finding the exponential parameter for that subset that yields the best
compression conditioned on the values on the boundary . Our
estimate is derived from a temporally stationary sequence of observations on
the set . We discuss how this method can also be applied to estimate a
spatially invariant parameter from a single configuration, and in so doing,
derive the Maximum Pseudo-Likelihood (MPL) estimate.Comment: Information Theory and Applications (ITA) workshop, February 201
Fuzzy transformations and extremality of Gibbs measures for the Potts model on a Cayley tree
We continue our study of the full set of translation-invariant splitting
Gibbs measures (TISGMs, translation-invariant tree-indexed Markov chains) for
the -state Potts model on a Cayley tree. In our previous work \cite{KRK} we
gave a full description of the TISGMs, and showed in particular that at
sufficiently low temperatures their number is .
In this paper we find some regions for the temperature parameter ensuring
that a given TISGM is (non-)extreme in the set of all Gibbs measures.
In particular we show the existence of a temperature interval for which there
are at least extremal TISGMs.
For the Cayley tree of order two we give explicit formulae and some numerical
values.Comment: 44 pages. To appear in Random Structures and Algorithm
Differential entropy and time
We give a detailed analysis of the Gibbs-type entropy notion and its
dynamical behavior in case of time-dependent continuous probability
distributions of varied origins: related to classical and quantum systems. The
purpose-dependent usage of conditional Kullback-Leibler and Gibbs (Shannon)
entropies is explained in case of non-equilibrium Smoluchowski processes. A
very different temporal behavior of Gibbs and Kullback entropies is confronted.
A specific conceptual niche is addressed, where quantum von Neumann, classical
Kullback-Leibler and Gibbs entropies can be consistently introduced as
information measures for the same physical system. If the dynamics of
probability densities is driven by the Schr\"{o}dinger picture wave-packet
evolution, Gibbs-type and related Fisher information functionals appear to
quantify nontrivial power transfer processes in the mean. This observation is
found to extend to classical dissipative processes and supports the view that
the Shannon entropy dynamics provides an insight into physically relevant
non-equilibrium phenomena, which are inaccessible in terms of the
Kullback-Leibler entropy and typically ignored in the literature.Comment: Final, unabridged version; http://www.mdpi.org/entropy/ Dedicated to
Professor Rafael Sorkin on his 60th birthda
Statistical Mechanics of Surjective Cellular Automata
Reversible cellular automata are seen as microscopic physical models, and
their states of macroscopic equilibrium are described using invariant
probability measures. We establish a connection between the invariance of Gibbs
measures and the conservation of additive quantities in surjective cellular
automata. Namely, we show that the simplex of shift-invariant Gibbs measures
associated to a Hamiltonian is invariant under a surjective cellular automaton
if and only if the cellular automaton conserves the Hamiltonian. A special case
is the (well-known) invariance of the uniform Bernoulli measure under
surjective cellular automata, which corresponds to the conservation of the
trivial Hamiltonian. As an application, we obtain results indicating the lack
of (non-trivial) Gibbs or Markov invariant measures for "sufficiently chaotic"
cellular automata. We discuss the relevance of the randomization property of
algebraic cellular automata to the problem of approach to macroscopic
equilibrium, and pose several open questions.
As an aside, a shift-invariant pre-image of a Gibbs measure under a
pre-injective factor map between shifts of finite type turns out to be always a
Gibbs measure. We provide a sufficient condition under which the image of a
Gibbs measure under a pre-injective factor map is not a Gibbs measure. We point
out a potential application of pre-injective factor maps as a tool in the study
of phase transitions in statistical mechanical models.Comment: 50 pages, 7 figure
On the small-scale structure of turbulence and its impact on the pressure field
Understanding the small-scale structure of incompressible turbulence and its
implications for the non-local pressure field is one of the fundamental
challenges in fluid mechanics. Intense velocity gradient structures tend to
cluster on a range of scales which affects the pressure through a Poisson
equation. Here we present a quantitative investigation of the spatial
distribution of these structures conditional on their intensity for
Taylor-based Reynolds numbers in the range [160, 380]. We find that the
correlation length, the second invariant of the velocity gradient, is
proportional to the Kolmogorov scale. It also is a good indicator for the
spatial localization of intense enstrophy and strain-dominated regions, as well
as the separation between them. We describe and quantify the differences in the
two-point statistics of these regions and the impact they have on the
non-locality of the pressure field as a function of the intensity of the
regions. Specifically, across the examined range of Reynolds numbers, the
pressure in strong rotation-dominated regions is governed by a
dissipation-scale neighbourhood. In strong strain-dominated regions, on the
other hand, it is determined primarily by a larger neighbourhood reaching
inertial scales.Comment: Accepted for publication by the Journal of Fluid Mechanic
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