655,529 research outputs found

### Generalised exponential families and associated entropy functions

A generalised notion of exponential families is introduced. It is based on
the variational principle, borrowed from statistical physics. It is shown that
inequivalent generalised entropy functions lead to distinct generalised
exponential families. The well-known result that the inequality of Cramer and
Rao becomes an equality in the case of an exponential family can be
generalised. However, this requires the introduction of escort probabilities.Comment: 20 page

### Information Equation of State

Landauer's principle is applied to information in the universe. Once stars
began forming, the increasing proportion of matter at high stellar temperatures
compensated for the expanding universe to provide a near constant information
energy density. The information equation of state was close to the dark energy
value, w = -1, for a wide range of redshifts, 10> z >0.8, over one half of
cosmic time. A reasonable universe information bit content of only 10^87 bits
is sufficient for information energy to account for all dark energy. A time
varying equation of state with a direct link between dark energy and matter,
and linked to star formation in particular, is clearly relevant to the cosmic
coincidence problem.In answering the "Why now?" question we wonder "What next?"
as we expect the information equation of state to tend towards w = 0 in the
future.Comment: 10 pages, 2 figure

### Poisson suspensions and entropy for infinite transformations

The Poisson entropy of an infinite-measure-preserving transformation is
defined as the Kolmogorov entropy of its Poisson suspension. In this article,
we relate Poisson entropy with other definitions of entropy for infinite
transformations: For quasi-finite transformations we prove that Poisson entropy
coincides with Krengel's and Parry's entropy. In particular, this implies that
for null-recurrent Markov chains, the usual formula for the entropy $-\sum q_i
p_{i,j}\log p_{i,j}$ holds in any of the definitions for entropy. Poisson
entropy dominates Parry's entropy in any conservative transformation. We also
prove that relative entropy (in the sense of Danilenko and Rudolph) coincides
with the relative Poisson entropy. Thus, for any factor of a conservative
transformation, difference of the Krengel's entropy is equal to the difference
of the Poisson entropies. In case there exists a factor with zero Poisson
entropy, we prove the existence of a maximum (Pinsker) factor with zero Poisson
entropy. Together with the preceding results, this answers affirmatively the
question raised in arXiv:0705.2148v3 about existence of a Pinsker factor in the
sense of Krengel for quasi-finite transformations.Comment: 25 pages, a final section with some more results and questions adde

### On Statistical Properties of Jizba-Arimitsu Hybrid Entropy

Jizba-Arimitsu entropy (also called hybrid entropy) combines axiomatics of
R\'enyi and Tsallis entropy. It has many common properties with them, on the
other hand, some aspects as e.g., MaxEnt distributions, are completely
different from the former two entropies. In this paper, we demonstrate the
statistical properties of hybrid entropy, including the definition of hybrid
entropy for continuous distributions, its relation to discrete entropy and
calculation of hybrid entropy for some well-known distributions. Additionally,
definition of hybrid divergence and its connection to Fisher metric is also
discussed. Interestingly, the main properties of continuous hybrid entropy and
hybrid divergence are completely different from measures based on R\'enyi and
Tsallis entropy. This motivates us to introduce average hybrid entropy, which
can be understood as an average between Tsallis and R\'enyi entropy

### Entropy rate of higher-dimensional cellular automata

We introduce the entropy rate of multidimensional cellular automata. This
number is invariant under shift-commuting isomorphisms; as opposed to the
entropy of such CA, it is always finite. The invariance property and the
finiteness of the entropy rate result from basic results about the entropy of
partitions of multidimensional cellular automata. We prove several results that
show that entropy rate of 2-dimensional automata preserve similar properties of
the entropy of one dimensional cellular automata.
In particular we establish an inequality which involves the entropy rate, the
radius of the cellular automaton and the entropy of the d-dimensional shift. We
also compute the entropy rate of permutative bi-dimensional cellular automata
and show that the finite value of the entropy rate (like the standard entropy
of for one-dimensional CA) depends on the number of permutative sites.
Finally we define the topological entropy rate and prove that it is an
invariant for topological shift-commuting conjugacy and establish some
relations between topological and measure-theoretic entropy rates

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