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Measurement Invariance, Entropy, and Probability

Steven A. Frank, D. Eric Smith
2010 Entropy  
Incorporating information about measurement scale into maximum entropy provides a general approach to the relations between measurement, information and probability.  ...  Our approach extends the method of maximum entropy to use measurement scale as a type of information constraint.  ...  Measurement, Information Invariance, and Probability We derive most likely probability distributions. Our method follows the maximum entropy approach [4, 7, 8] .  ... 
doi:10.3390/e12030289 fatcat:g3a2wla3azh33crgenlhydctje

Φ-Entropy Inequality and Invariant Probability Measure for SDEs with Jump [article]

Feng-Yu Wang
2013 arXiv   pre-print
The semigroup Φ-entropy inequality for SDEs driven by Poisson point processes as well as a sharp result on the existence of invariant probability measures are also presented.  ...  By using the Φ-entropy inequality derived in Wu, Ch for Poisson measures, the same type of inequality is established for a class of stochastic differential equations driven by purely jump Lévy processes  ...  Next, partly for the proof of Theorem 1.1(2), we consider the existence of invariant probability measures for the following more general SDE: Although the existence of invariant probability measures for  ... 
arXiv:1309.0942v2 fatcat:wsjpv43qsvfjxdhj3r2rmvcoya

Distribution-Invariant Risk Measures, Entropy, and Large Deviations

Stefan Weber
2007 Journal of Applied Probability  
For distribution-invariant risk measures which are continuous on compacts, we employ the theory of large deviations to study the probability of large errors.  ...  If the approximate risk measurements are based on the empirical distribution of independent samples, then the rate function equals the minimal relative entropy under a risk measure constraint.  ...  Acknowledgements I would like to thank Boris Buchmann, Hans Föllmer, and Ulrich Horst for helpful discussions. I am very grateful for the useful comments of an anonymous referee.  ... 
doi:10.1017/s0021900200002692 fatcat:at6ck5evx5f2zp7tpyj5qmrohq

Distribution-Invariant Risk Measures, Entropy, and Large Deviations

Stefan Weber
2007 Journal of Applied Probability  
For distribution-invariant risk measures which are continuous on compacts, we employ the theory of large deviations to study the probability of large errors.  ...  If the approximate risk measurements are based on the empirical distribution of independent samples, then the rate function equals the minimal relative entropy under a risk measure constraint.  ...  Acknowledgements I would like to thank Boris Buchmann, Hans Föllmer, and Ulrich Horst for helpful discussions. I am very grateful for the useful comments of an anonymous referee.  ... 
doi:10.1239/jap/1175267161 fatcat:6jron4zkizfa7byblbejh7qsue

Invariant probability and entropy measures for multidimensional maps of chaotic attractors

R. Riganti, M.G. Zavattaro
1996 Computers and Mathematics with Applications  
we define and calculate the probability density in phase spsce and the information entropy of Poincare' maps describing the chaotic, stationary behaviour of deterministic dynamical systems with many degrees  ...  By partitioning the phsse spsce into cells and computing the probability of finding points of the map in each cell, a numeric approximation of the two statistical measures is derived, and the results are  ...  INVARIANT PROBABILITY DENSITY AND ENTROPY OF POINCARE' MAPS Given a Poincare' map representing an attractor of a dynamical system in a volume V of a &dimensional phase space, let US cover V by a number  ... 
doi:10.1016/s0898-1221(96)00182-4 fatcat:yfeq3iflirde7h7pdywzb3h4fa

Probability distribution and entropy as a measure of uncertainty

Qiuping A Wang
2008 Journal of Physics A: Mathematical and Theoretical  
The relationship between three probability distributions and their optimal entropy forms is discussed without postulating entropy property as usual.  ...  For this purpose, the entropy I is defined as a measure of uncertainty of the probability distribution p(x) of a random variable x by a variational relationship dx , a definition assuring the optimization  ...  Tsobnang and Professor S. Abe for very useful discussion and comments.  ... 
doi:10.1088/1751-8113/41/6/065004 fatcat:rqhxauojqzfhbfsbpcxrdqvnmm

Predictability, topological entropy and invariant random orders [article]

Andrei Alpeev, Tom Meyerovitch, Sieye Ryu
2019 arXiv   pre-print
On route, we investigate invariant random orders and formulate a unified Kieffer-Pinsker formula for the Kolmogorov-Sinai entropy of measure preserving actions of amenable groups.  ...  We prove that a topologically predictable action of a countable amenable group has zero topological entropy, as conjectured by Hochman.  ...  We will denote by Prob(X) the (compact, convex) space of Borel probability measures on X, and by Prob Γ (X) the subset of Γ-invariant Borel probability measures.  ... 
arXiv:1812.10833v2 fatcat:7xikjnaz3bc5fco5sqy45ishia

Entropy computing via integration over fractal measures

Wojciech Słomczyński, Jarosław Kwapień, Karol Życzkowski
2000 Chaos  
We discuss the properties of invariant measures corresponding to iterated function systems (IFSs) with place-dependent probabilities and compute their Renyi entropies, generalized dimensions, and multifractal  ...  This provides a new method of computing entropy for some classical and quantum dynamical systems. Numerical techniques are based on integration over the fractal measures.  ...  It is easy to prove that the IFS F q satisfies our general assumption, and hence, a unique invariant probability measure µ q exists.  ... 
doi:10.1063/1.166492 pmid:12779373 fatcat:2oi23gbwvrgpzoadpmgpy2iflu

Invariance in ecological pattern [article]

Steven A. Frank, Jordi Bascompte
2019 arXiv   pre-print
First, invariance provides a simpler and more fundamental maximum entropy derivation of the log series distribution.  ...  Common answers include maximum entropy, neutrality, and convergent outcome from different underlying biological processes.  ...  Acknowledgments The Donald Bren Foundation (SAF) and the Swiss National Science Foundation (grant 31003A_169671 to JB) support our research.  ... 
arXiv:1906.06979v2 fatcat:slyucxhcq5cxhfu4ro5blsciom

Invariance in ecological pattern

Steven A. Frank, Jordi Bascompte
2019 F1000Research  
First, invariance provides a simpler and more fundamental maximum entropy derivation of the log series distribution.  ...  Common answers include maximum entropy, neutrality, and convergent outcome from different underlying biological processes.  ...  That scale provided a much simpler analysis, in which r is the incremental scale with respect to invariance and the measurement scale with respect to entropy.  ... 
doi:10.12688/f1000research.21586.1 pmid:32089829 pmcid:PMC7008606 fatcat:vjkikqnpwnbtzl7iemx6rsqukm

Invariance in ecological pattern [article]

Steven A. Frank, Jordi Bascompte
2019 bioRxiv   pre-print
First, invariance provides a simpler and more fundamental maximum entropy derivation of the log series distribution.  ...  Common answers include maximum entropy, neutrality, and convergent outcome from different underlying biological processes.  ...  Acknowledgments The Donald Bren Foundation (SAF) and the Swiss National Science Foundation (grant 31003A_169671 to JB) support our research.  ... 
doi:10.1101/673590 fatcat:mluuk3nnnvedrbrotwc4rsflju

Self-deprivation: A review

Robert A. Frank, Robert M. Stutz
1984 Psychological bulletin  
Particular biological processes set the relation between the information in observations and magnitude, the basis for information invariance, symmetry and measurement scale.  ...  The measurement scale, in turn, determines the most likely probability distributions and observed patterns associated with particular processes.  ...  Intuitive description of measurement and probability Intuitively, one can think of the symmetry of information invariance and measurement scale in the following way.  ... 
doi:10.1037/0033-2909.96.2.384 fatcat:o5me5fk7ejf53fgpn2m4od4bii

Self-deprivation: A review

Robert A. Frank, Robert M. Stutz
1984 Psychological bulletin  
Particular biological processes set the relation between the information in observations and magnitude, the basis for information invariance, symmetry and measurement scale.  ...  The measurement scale, in turn, determines the most likely probability distributions and observed patterns associated with particular processes.  ...  Intuitive description of measurement and probability Intuitively, one can think of the symmetry of information invariance and measurement scale in the following way.  ... 
doi:10.1037//0033-2909.96.2.384 fatcat:wm4sai47ezcbjfwr6kp5642zue

A simple derivation and classification of common probability distributions based on information symmetry and measurement scale

S. A. FRANK, E. SMITH
2011 Journal of Evolutionary Biology  
Second, different problems associate magnitude to information in different ways, an association we describe in terms of the relation between information invariance and measurement scale.  ...  Our framework relates the different continuous probability distributions through the variations in measurement scale that change each family of maximum entropy distributions into a distinct family.  ...  Intuitive description of measurement and probability Intuitively, one can think of the symmetry of information invariance and measurement scale in the following way.  ... 
doi:10.1111/j.1420-9101.2010.02204.x pmid:21265914 fatcat:bkdluwvmuvfcvn5v3ud4anczay

Ruelle's inequality in negative curvature

Felipe Riquelme
2018 Discrete and Continuous Dynamical Systems. Series A  
As an application, we prove Ruelle's inequality and Pesin's entropy formula for the geodesic flow in manifolds with pinched negative sectional curvature.  ...  In this paper we study different notions of entropy for measurepreserving dynamical systems defined on noncompact spaces.  ...  I would also like to thank Samuel Tapie for his inspiring remarks on some results of this paper, Godofredo Iommi for some nice suggestions that improved the presentation of this work, and the referee of  ... 
doi:10.3934/dcds.2018119 fatcat:fcgcvvlisbhwzdnew2dokb7tfm
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