31,285 research outputs found
Origins of the Combinatorial Basis of Entropy
The combinatorial basis of entropy, given by Boltzmann, can be written , where is the dimensionless entropy, is the
number of entities and is number of ways in which a given
realization of a system can occur (its statistical weight). This can be
broadened to give generalized combinatorial (or probabilistic) definitions of
entropy and cross-entropy: and , where is the probability of a given
realization, is a convenient transformation function, is a
scaling parameter and an arbitrary constant. If or
satisfy the multinomial weight or distribution, then using
and , and asymptotically
converge to the Shannon and Kullback-Leibler functions. In general, however,
or need not be multinomial, nor may they approach an
asymptotic limit. In such cases, the entropy or cross-entropy function can be
{\it defined} so that its extremization ("MaxEnt'' or "MinXEnt"), subject to
the constraints, gives the ``most probable'' (``MaxProb'') realization of the
system. This gives a probabilistic basis for MaxEnt and MinXEnt, independent of
any information-theoretic justification.
This work examines the origins of the governing distribution ....
(truncated)Comment: MaxEnt07 manuscript, version 4 revise
Combinatorial Information Theory: I. Philosophical Basis of Cross-Entropy and Entropy
This study critically analyses the information-theoretic, axiomatic and
combinatorial philosophical bases of the entropy and cross-entropy concepts.
The combinatorial basis is shown to be the most fundamental (most primitive) of
these three bases, since it gives (i) a derivation for the Kullback-Leibler
cross-entropy and Shannon entropy functions, as simplified forms of the
multinomial distribution subject to the Stirling approximation; (ii) an
explanation for the need to maximize entropy (or minimize cross-entropy) to
find the most probable realization; and (iii) new, generalized definitions of
entropy and cross-entropy - supersets of the Boltzmann principle - applicable
to non-multinomial systems. The combinatorial basis is therefore of much
broader scope, with far greater power of application, than the
information-theoretic and axiomatic bases. The generalized definitions underpin
a new discipline of ``{\it combinatorial information theory}'', for the
analysis of probabilistic systems of any type.
Jaynes' generic formulation of statistical mechanics for multinomial systems
is re-examined in light of the combinatorial approach. (abbreviated abstract)Comment: 45 pp; 1 figure; REVTex; updated version 5 (incremental changes
Measure Recognition Problem
This is an article in mathematics, specifically in set theory. On the example
of the Measure Recognition Problem (MRP) the article highlights the phenomenon
of the utility of a multidisciplinary mathematical approach to a single
mathematical problem, in particular the value of a set-theoretic analysis. MRP
asks if for a given Boolean algebra \algB and a property of measures
one can recognize by purely combinatorial means if \algB supports a strictly
positive measure with property . The most famous instance of this problem
is MRP(countable additivity), and in the first part of the article we survey
the known results on this and some other problems. We show how these results
naturally lead to asking about two other specific instances of the problem MRP,
namely MRP(nonatomic) and MRP(separable). Then we show how our recent work D\v
zamonja and Plebanek (2006) gives an easy solution to the former of these
problems, and gives some partial information about the latter. The long term
goal of this line of research is to obtain a structure theory of Boolean
algebras that support a finitely additive strictly positive measure, along the
lines of Maharam theorem which gives such a structure theorem for measure
algebras
Game-theoretic Resource Allocation Methods for Device-to-Device (D2D) Communication
Device-to-device (D2D) communication underlaying cellular networks allows
mobile devices such as smartphones and tablets to use the licensed spectrum
allocated to cellular services for direct peer-to-peer transmission. D2D
communication can use either one-hop transmission (i.e., in D2D direct
communication) or multi-hop cluster-based transmission (i.e., in D2D local area
networks). The D2D devices can compete or cooperate with each other to reuse
the radio resources in D2D networks. Therefore, resource allocation and access
for D2D communication can be treated as games. The theories behind these games
provide a variety of mathematical tools to effectively model and analyze the
individual or group behaviors of D2D users. In addition, game models can
provide distributed solutions to the resource allocation problems for D2D
communication. The aim of this article is to demonstrate the applications of
game-theoretic models to study the radio resource allocation issues in D2D
communication. The article also outlines several key open research directions.Comment: Accepted. IEEE Wireless Comms Mag. 201
Adaptive group testing as channel coding with feedback
Group testing is the combinatorial problem of identifying the defective items
in a population by grouping items into test pools. Recently, nonadaptive group
testing - where all the test pools must be decided on at the start - has been
studied from an information theory point of view. Using techniques from channel
coding, upper and lower bounds have been given on the number of tests required
to accurately recover the defective set, even when the test outcomes can be
noisy.
In this paper, we give the first information theoretic result on adaptive
group testing - where the outcome of previous tests can influence the makeup of
future tests. We show that adaptive testing does not help much, as the number
of tests required obeys the same lower bound as nonadaptive testing. Our proof
uses similar techniques to the proof that feedback does not improve channel
capacity.Comment: 4 pages, 1 figur
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