7,634 research outputs found
The Blackwell relation defines no lattice
Blackwell's theorem shows the equivalence of two preorders on the set of
information channels. Here, we restate, and slightly generalize, his result in
terms of random variables. Furthermore, we prove that the corresponding partial
order is not a lattice; that is, least upper bounds and greatest lower bounds
do not exist.Comment: 5 pages, 1 figur
Sharp and fuzzy observables on effect algebras
Observables on effect algebras and their fuzzy versions obtained by means of
confidence measures (Markov kernels) are studied. It is shown that, on effect
algebras with the (E)-property, given an observable and a confidence measure,
there exists a fuzzy version of the observable. Ordering of observables
according to their fuzzy properties is introduced, and some minimality
conditions with respect to this ordering are found. Applications of some
results of classical theory of experiments are considered.Comment: 23 page
Canonical extensions and ultraproducts of polarities
J{\'o}nsson and Tarski's notion of the perfect extension of a Boolean algebra
with operators has evolved into an extensive theory of canonical extensions of
lattice-based algebras. After reviewing this evolution we make two
contributions. First it is shown that the failure of a variety of algebras to
be closed under canonical extensions is witnessed by a particular one of its
free algebras. The size of the set of generators of this algebra can be made a
function of a collection of varieties and is a kind of Hanf number for
canonical closure. Secondly we study the complete lattice of stable subsets of
a polarity structure, and show that if a class of polarities is closed under
ultraproducts, then its stable set lattices generate a variety that is closed
under canonical extensions. This generalises an earlier result of the author
about generation of canonically closed varieties of Boolean algebras with
operators, which was in turn an abstraction of the result that a first-order
definable class of Kripke frames determines a modal logic that is valid in its
so-called canonical frames
A universal approach for drainage basins
Drainage basins are essential to Geohydrology and Biodiversity. Defining
those regions in a simple, robust and efficient way is a constant challenge in
Earth Science. Here, we introduce a model to delineate multiple drainage basins
through an extension of the Invasion Percolation-Based Algorithm (IPBA). In
order to prove the potential of our approach, we apply it to real and
artificial datasets. We observe that the perimeter and area distributions of
basins and anti-basins display long tails extending over several orders of
magnitude and following approximately power-law behaviors. Moreover, the
exponents of these power laws depend on spatial correlations and are invariant
under the landscape orientation, not only for terrestrial, but lunar and
martian landscapes. The terrestrial and martian results are statistically
identical, which suggests that a hypothetical martian river would present
similarity to the terrestrial rivers. Finally, we propose a theoretical value
for the Hack's exponent based on the fractal dimension of watersheds,
. We measure for Earth, which is close to
our estimation of . Our study suggests that Hack's law can
have its origin purely in the maximum and minimum lines of the landscapes.Comment: 20 pages, 6 Figures, and 1 Tabl
N-fold way simulated tempering for pairwise interaction point processes
Pairwise interaction point processes with strong interaction are usually difficult to
sample. We discuss how Besag lattice processes can be used in a simulated tempering
MCMC scheme to help with the simulation of such processes. We show how
the N-fold way algorithm can be used to sample the lattice processes efficiently
and introduce the N-fold way algorithm into our simulated tempering scheme. To
calibrate the simulated tempering scheme we use the Wang-Landau algorithm
Unique Information and Secret Key Decompositions
The unique information () is an information measure that quantifies a
deviation from the Blackwell order. We have recently shown that this quantity
is an upper bound on the one-way secret key rate. In this paper, we prove a
triangle inequality for the , which implies that the is never greater
than one of the best known upper bounds on the two-way secret key rate. We
conjecture that the lower bounds the two-way rate and discuss implications
of the conjecture.Comment: 7 page
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